Borehole correction system for an array induction well-logging apparatus

A new borehole correction software, for use in a well logging truck computer, obtains an accurate measurement of the true conductivity of a formation in a borehole by subtracting a correction term, which is a function of the mud conductivity, the borehole radius, the standoff distance, and an estimated value of the formation conductivity, from certain raw data received by a receiver thereby producing a set of eighteen complex output voltages which represent signals that would have been recorded from the receiver had there been no borehole.

BACKGROUND OF THE INVENTION 
The invention of the subject application pertains to an array induction 
logging apparatus for oil well boreholes and more particularly, to a 
borehole correction system associated with the induction logging apparatus 
for correcting for the effects of the borehole on an overall formation 
conductivity measurement. 
An induction logging apparatus, disposed in a wellbore, or borehole of an 
oil well, basically comprises at least one transmitting coil and at least 
one receiving coil mounted on a support and axially spaced from each other 
in the direction of the borehole. The transmitting coil is energized by an 
alternating current at a frequency which is typically 20 kHz and generates 
an electric field which induces in the formation, surrounding the 
borehole, eddy currents which flow coaxially to the borehole and the 
intensity of which is proportional to the conductivity of the formation. 
The field generated in turn by these eddy currents induces in the 
receiving coil an electromotive force (EMF), which produces a received 
signal in the receiving coil. By suitably processing the received signal 
from the receiving coil, a measurement of the conductivity of the 
formation is obtained. 
However the conductivity of the mud or drilling fluid in the borehole may 
distort the measured value of the conductivity of the formation. One prior 
art induction logging apparatus, known as the 6FF40 sonde, consists of 
three transmitter coils and three receiver coils. In the 6FF40 sonde, the 
effective coil spacing is about 40 inches; the coil positions and the 
number of turns on the coils is designed to minimize the effect of the 
borehole. Therefore, although the conductivity of the mud (sigma.sub.m or 
s.sub.m) did distort the determination of the true formation conductivity 
(Sigma.sub.t or s.sub.f) the distortion was small and required a 
relatively minor correction. As a result, with the aforementioned prior 
art well logging apparatus, the borehole corrections could be applied 
after recording and signal processing the data, and frequently were left 
up to the formation analyst interpreting the induction logs. However, a 
new well logging array induction tool was developed, the characteristics 
of which were disclosed in prior pending application Ser. No. 043,130, 
filed Apr. 27, 1987 entitled "Induction Logging Method and Apparatus", and 
in prior pending application Ser. No. 932,231, filed Nov. 18, 1986 
entitled "Induction Logging Sonde With Metallic Support", now U.S. Pat. 
No. 4,873,488. These applications, Ser. Nos. 043,130 and 932,231, are 
incorporated by reference into the specification of this application. This 
new well logging array induction tool (hereinafter termed the "AIT Tool") 
has one transmitter coil and nine (9) receivers with two coils per 
receiver. The spacing between transmitter and receiver coil pairs ranges 
from 6 inches to 72 inches. The signals from the nine receivers collect 
information about the conductivity at different depths in the formation. 
Because there are so many more measurements in the new AIT tool, relative 
to the prior art induction logging apparatus, manual correction of the 
signal for the effect of the borehole and would be cumbersome and 
time-consuming. Also in the AIT the receivers with a small spacing are 
more strongly affected by the conductive borehole fluid, and the 
correction is not a small fraction of the recorded signal. As a result, 
using the new AIT tool, in order to obtain an accurate measurement of the 
true conductivity of the formation s.sub.t, the measured EMF for each 
receiver must be corrected for the effect of the borehole in software, 
prior to signal processing. The portion of the induced EMF associated with 
the conductivity of the mud, s.sub.m, must be removed from the induced EMF 
in each of the nine receivers thereby yielding an EMF which represents the 
actual conductivity of the formation. The induced EMF signals form the 
AIT, after applying borehole corrections, and after signal processing, 
give an indication of the conductivity in the formation at different 
radial distances from the borehole. When the AIT induction logs are 
combined with other information such as the porosity of the rock, and the 
ground water conductivity, they can be used to infer the water saturation 
which indicates the presence of hydrocarbons. Because the AIT probes 
different radial distances into the formation, it will also indicate 
invasion of the formation rock by fluids from the borehole. 
A prior art technique for correcting an induction log for the effect of the 
borehole is set forth in a publication entitled "Phasor Induction Tool", 
published by the assignee of this application, dated July 1986. A similar 
prior art technique is set forth in another publication entitled 
"Schlumberger Log Interpretation Chart", the 1979 edition. In both the 
aforementioned publications, the described prior art technique for 
performing the borehole correction involves manually performing the 
subtraction by referring to set of curves (borehole geometrical factor 
versus hole diameter), calculating the contribution to the induced EMF 
associated with the conductivity of the mud, and performing the 
aforementioned subtraction. 
This prior art manual technique would be very cumbersome for an array 
induction sonde having many receiving coils, such as the new AIT tool. 
Furthermore it would not be sufficiently accurate for the signals coming 
from the short-spacing receivers. 
SUMMARY OF THE INVENTION 
Accordingly, it is a primary object of the present invention to obtain an 
accurate measurement of the true conductivity of a formation, s.sub.t, at 
different depths and at different radial distances surrounding a borehole 
of an oil well with an induction sonde consisting of an array of receiver 
coils. 
It is another object of the present invention to obtain an accurate 
measurement of the true conductivity of a formation, s.sub.t, by applying 
new borehole correction software for use with a well-site computer, the 
software automatically correcting the measured value of the induced EMF in 
each receiver by removing therefrom the contribution associated with the 
conductivity of the mud in the borehole thereby yielding voltage values 
which represent the true conductivity of the formation s.sub.t. 
These and other objects of the present invention are accomplished by 
developing borehole correction software which functions in association 
with a well-site computer connected to the new AIT array induction logging 
tool. The transmitter coil in the AIT carries alternating currents of four 
different frequencies which generate electric fields in the formation 
surrounding the borehole, which electric fields induce eddy currents 
proportional to the conductivity in the formation. Electromotive forces 
are induced in nine different receiver coil pairs. In each receiver, the 
open-circuit voltage at two different frequencies is detected, amplified, 
and converted to digital form. The open-circuit voltage is represented as 
a complex number, the real part of which is the component of the voltage 
in phase with the transmitter current, and the imaginary part of which is 
the quadrature (90-degree out-of-phase) component of the open-circuit 
voltage Because two frequencies are detected in each receiver, the data 
recorded for each vertical depth of the AIT consists of eighteen (18) 
complex voltages, denoted V.sub.meas. These eighteen signals (otherwise 
termed "AIT Raw Data") are processed in the wellsite computer by the 
borehole correction software of the present invention using, in addition 
to the AIT raw data, certain other borehole or wellbore parameters, such 
as the conductivity of the mud sm, the radius of the borehole or wellbore 
r, and the tool stand of distance x, that is, the distance between the AIT 
tool external surface and the borehole wall (in the off-center case) An 
AIT log can be operated with the tool centered in the borehole by means of 
centralizers, or off-center by using rubber fins to maintain a constant 
standoff distance. The AIT borehole correction software has provisions for 
both model of operation. The borehole correction software processes the 
AIT raw data b removing therefrom a correction term which is a function of 
the mud conductivity s.sub.m, the borehole radius r, the standoff distance 
x, and an estimated value of -the formation conductivity s.sub.f. The 
output of the borehole correction software is a set of eighteen complex 
voltages, denoted V.sub.corr which represent the signals that would have 
been recorded in the absence of a borehole. 
The borehole correction software processes the input data stream of AIT raw 
signals V.sub.meas and produces an output data stream V.sub.corr by using 
a relation of the form 
EQU V.sub.corr =V.sub.meas -V.sub.model (s.sub.m,s.sub.f,r,x) . 
Here V.sub.model is derived from model calculations, and depends on four 
parameters: the mud conductivity s.sub.m, an estimated formation 
conductivity s.sub.f, the borehole radius r, and the standoff distance x. 
Several options are available for selecting the model parameters. The mud 
conductivity s.sub.m is obtained from a mud resistivity logging apparatus 
if available or, if not available, from a sample of the borehole fluid. 
The formation conductivity estimate s.sub.f can be obtained from a shallow 
resistivity log such as a micro-SFL (micro spherically focused log) if 
available, or, if not available by performing a least square minimization 
using the V.sub.meas data for the shortspacing arrays. The borehole radius 
r can be obtained from a borehole caliper tool if available, or, if not 
available, can be estimated from the V.sub.meas data by performing a least 
squares minimization. For the mud standoff distance x, one can use the 
nominal standoff distance determined by the standoff fins mounted on the 
body of the AIT tool, or, for an irregular borehole, an effective standoff 
distance can be estimated from the V.sub.meas preforming a least squares 
minimization. The Levenberg-Marquard method is used to determine optimum 
values for the parameters that are being estimated by the least squares 
criterion. 
Further scope of applicability of the present invention will become 
apparent from the detailed description presented hereinafter. It should be 
understood, however, that the detailed description and the specific 
examples, while representing a preferred embodiment of the invention, are 
given by way of illustration only, since various changes and modifications 
within the spirit and scope of the invention will become obvious to one 
skilled in the art from a reading of the following detailed description.

DESCRIPTION OF THE PREFERRED EMBODIMENT 
Referring to FIG. 1, a borehole of an oil well is illustrated. An array 
induction tool (AIT) 10 is disposed in the borehole, the AIT tool 10 being 
connected by a wireline cable to a well-logging truck at the surface of 
the well. The well-logging truck contains a computer in which the borehole 
correction software of the present invention is stored. The well-logging 
truck computer may comprise any typical computer, such as the computer set 
forth in U.S. Pat. No. 4,713,751 entitled "Masking Commands for a Second 
Processor When a First Processor Requires a Flushing Operation in a 
Multiprocessor System", the disclosure of which is incorporated by 
reference into the specification of this application. The AIT tool 
comprises one transmitter coil 12 and nine (9) receivers 18 each 
consisting of two coils. The transmitter coil is energized by a known 
reference current consisting of four frequencies, approximately 25, 50, 
100, and 200 kHz (kiloHerz). The transmitter coil generates an electric 
field in the formation, which produces a coaxial eddy current proportional 
to the formation conductivity which then induces an electromotive force in 
each of the 9 receivers. Each receiver 18 comprises a main coil and a 
bucking coil connected in series. 
Referring to FIG. 2, a more detailed construction of the AIT tool 10, of 
FIG. 1, is illustrated. The receiver coils respond with a voltage signal 
V1 which depends on formation characteristics. The voltage signal V1 is 
amplified, filtered into four separate frequencies, and resolved into 
in-phase and quadrature components, by the circuitry 20, 22. The voltage 
is represented in complex (phasor) notation with the in-phase component as 
the real part of the complex number, and the quadrature (90-degree 
out-of-phase) component as the imaginary part of the complex number. Two 
frequencies are selected for each of the nine arrays. Thus for each depth 
of the AIT tool in the well, eighteen (18) complex voltages are recorded. 
This data set will be referred to as "AIT raw data". The AIT raw data are 
transmitted by telemetry to the logging truck computer at the surface for 
processing. 
The AIT tool 10 of FIG. 2 is set forth in detail in prior pending 
application Ser. No. 043,130, filed Apr. 27, 1987, the disclosure of which 
is incorporated by reference into the specification of this application. 
In FIG. 2, simple individual arrays are preferred generally consisting of a 
single transmitter coil 12 and two receiver coils 16, 14 (16', 14'). The 
basic three-coil induction sensor includes a primary receiver coil 16 
located at a distance L from the transmitter. A secondary or bucking coil 
14 acts as a mutual inductance balancing coil. The secondary coil is 
connected in series with the primary coil, but is wound in a opposite 
sense to the primary coil. The placement of the secondary coil between the 
transmitter coil 12 and the primary receiver coil 16 is a matter of 
choice, but once its placement is fixed, the number of its windings may be 
selected so as substantially to balance or null the direct mutual coupling 
between the transmitter and the receiver array. If the position of the 
bucking coil is selected to be 3L/4 or three-fourths of the distance 
between the transmitter coil 12 and the primary receiver coil 16, the 
number of turns in the bucking coil should be approximately (0.75).sup.3= 
0.422 times the number of turns in the primary coil in order to achieve 
the balanced condition. 
Preferably the AIT tool 10 is constructed of a number of these simple 
arrays by placing a single transmitter 12 at the center of the tool and 
placing pairs of receiver coils such as pairs 16, 14 and 16', 14' on 
either side of it. Amplifiers 20, 20' and phase-sensitive detectors 22, 
22' (PSD) may be constructed of conventional analog induction electronics. 
As illustrated, a multi-frequency oscillator 26, operating at four 
frequencies 25, 50, 100 and 200 kHz, excite transmitter 12. The receiver 
arrays, spaced, for example, three feet from transmitter 12, respond with 
voltage signals, V.sub.1, V.sub.1 ' which depend on formation 
characteristics. Such voltage signals are amplified, filtered into 
frequency components at 25 and 50 kHz, and resolved into in-phase and 
quadrature (90-degree out-of-phase) components. 
In FIG. 3, the simple construction of the well-logging truck computer is 
illustrated. The computer comprises a processor 30, a tape drive, and main 
memory 40. The main memory 40 stores a set of software termed the 
"borehole correction software" 40a of the present invention. The computer 
of FIG. 3 may be any typical computer, such as the multiprocessor computer 
described in U.S. Pat. No. 4,713,751, referenced hereinabove, the 
disclosure of which is incorporated by reference into the specification of 
this application. 
In FIG. 4, a flow diagram of the borehole correction software, stored in 
the main memory 40 of FIG. 3, is illustrated. 
In FIG. 4, the borehole correction software 40a comprises an interface 40a1 
which receives the 18 complex voltages which comprise the AIT Raw Data and 
a set of borehole parameters. Recall that the AIT Raw Data includes the 
eighteen (18) complex numbers output from the AIT tool 10 disposed in the 
wellbore. The borehole parameters include the conductivity of mud 
(s.sub.m), the radius of the borehole (r), and the tool standoff distance 
(x), that is, the distance between the AIT tool 10 exterior surface and 
the borehole wall (in case the tool is operated off-center). The borehole 
parameters are shown more clearly in FIG. 5 of the drawings. The borehole 
correction software 40a further comprises a data buffer 40a2, connected to 
the interface via a driver 40a3, the data buffer 40a2 receiving the 
eighteen (18) complex numbers from the interface 40a1; a solver 40a4 which 
receives from the driver 40a3 given or fixed values of the conductivity of 
mud (s.sub.m) and the standoff distance (x), and determines, by least 
squares optimization, appropriate values for the other two parameters, the 
formation conductivity (s.sub.f), and the borehole radius (r); a 
correction module 40a5 which receives four borehole parameters s.sub.f, 
sm, r, and x, and computes model values for eight (8) of the eighteen (18) 
complex voltages that are a function of the four borehole parameters, the 
solver 40a4 receiving the eight model voltages from the correction module 
40a5 and the corresponding eight Raw Data measured voltages from the data 
buffer 40a2, and performing a least squares minimization to determine 
values of s.sub.f and r that most nearly match the eight measured complex 
voltages from the data buffer 40a2, the above minimization being performed 
iteratively until a final set of borehole parameters are determined; and a 
final correction module 40a6 which receives the final set of borehole 
parameters from the solver 40a4, and, after one call to the correction 
module 40a7 to compute model values for all eighteen (18) complex 
voltages, generates a set of borehole corrected data, the borehole 
corrected data representing an approximation to the voltages that the AIT 
would have recorded in the absence of the borehole. 
Referring to FIG. 6, the borehole correction software 40a call structure is 
illustrated in FIG. 6, the subroutines which comprise the interface 40a1, 
the solver 40a4, the driver 40a3, and the correction modules 40a5 and 40a7 
are illustrated. Furthermore, the call structure sequence is illustrated. 
For example, the driver 40a3 (BORCOR) calls each of the subroutines within 
the correction modules 40a5, 40a7, with the exception of subroutine 
DSMFEVAL, which subroutine is called by subroutine FCN in the solver 40a4. 
A detailed description of the borehole correction software 40a of FIGS. 3, 
4, and 6 is set forth in APPENDIX A provided hereinbelow. 
A functional description of the borehole correction software of the present 
invention will be set forth in the following paragraphs with reference to 
FIGS. 1-5 of the drawings, and, in particular, FIG. 4. This functional 
description will concentrate on one operating option of the borehole 
correction software, namely the case where the mud conductivity (s.sub.m) 
and the standoff (x) are known to the operating engineer, but the borehole 
radius (r) and an effective formation conductivity (s.sub.f) are not 
known, for example because a mechanical caliper is not available and/or a 
shallow resistivity log is not available. Options are available to allow 
the borehole correction software to select optimum values of other 
combinations of the model borehole parameters, for example the effective 
formation conductivity (s.sub.f) and the standoff (x). 
The AIT raw data from receiver coils 18 is received by the interface 40a1, 
the interface 40a1 transmitting the AIT raw data to the data buffer 40a2 
via driver 40a3 for temporary storage therein. Recall that the AIT raw 
data includes eighteen (18) complex voltages (two frequencies for each of 
the nine receiver coils) associated with each position occupied by the AIT 
tool 10 in the wellbore. For the operating option under consideration, the 
interface 40a1 also receives two borehole parameters which consist of the 
mud conductivity (s.sub.m) and the standoff distance (x), in the 
off-center case. The other two borehole parameters, the conductivity of 
the formation (s.sub.f) and the radius of the borehole r, are yet to be 
determined. The borehole parameters s.sub.m, and x are passed directly 
from interface 40a1 to the solver 40a4. 
The AIT raw data (eighteen complex voltages) V.sub.meas are transmitted to 
the driver module 40a3 and stored in the data buffer 40a2. The data buffer 
stores forty (40) depth samples for each of the 18 channels, so that 
borehole irregularities can be correlated in depth. The driver program 
then selects eight of the eighteen complex voltages V.sub.meas from 
appropriate depths and transmits these eight complex voltages to the 
solver 40a5. The eight complex voltages selected are those associated with 
the four receivers which are nearest the transmitter coil 12. The solver 
40a4 also receives two borehole parameters s.sub.m, x from the driver 
40a3, and selects an initial set of values for the remaining two borehole 
parameters s.sub.f, r. This first selected set of borehole parameters is 
passed to the correction module 40a5. 
In response thereto, the correction module 40a5 computes model values 
V.sub.model for the eight complex voltages that are a function of the 
initial set of borehole parameters s.sub.m, s.sub.f, r,x. The model values 
are computed from approximate formulas obtained by fitting tabulated 
values of the borehole signal; the tabulated values having been obtained 
by solving Maxwell's equations on a large mainframe computer. Listed below 
is the form of the in-phase component of the voltage for one channel (the 
real part of V.sub.model): 
##EQU1## 
where w is the following ratio: 
EQU w=(s.sub.m -s.sub.f)/(s.sub.m +s.sub.f). 
The functions c.sub.1, . . . , c.sub.8 have the following form: 
EQU c.sub.i (s.sub.f,r,x)=b.sub.i0 (r,x)+b.sub.il (r,x)+b.sub.i2 
(r,x)sqrt(s.sub.f) +b.sub.i3 (r,x)log(s.sub.f)+b.sub.i4 
(r,x)[log(s.sub.f)].sup.2, 
and the functions (b) are specified as follows: 
##EQU2## 
The coefficients (a) are constants, and have different numerical values 
for each receiver. In the above expressions log denotes the natural 
logarithm, and sqrt the square root. An expression similar to 
R(s.sub.m,s.sub.f,r,x) is used to compute the imaginary part (quadrature 
component) of V.sub.model. In the correction module, 40a7 in FIG. 6, the 
subroutines VDM06D, . . . , VDM72C evaluate expressions of this type to 
compute V.sub.model for each receiver and each frequency. 
The model's eight complex numbers Vj.sub.model, j=1, . . . , 8, are 
transmitted back to the solver 40a4. Recall that the eight complex numbers 
Vj.sub.meas, j=1, . . . , 8, were already passed directly to the solver 
40a4. The solver 40a4 compares V.sub.model with V.sub.meas to determine if 
V.sub.model most nearly matches V.sub.meas, the solver 40a4 using the 
following least squares criterion: 
##EQU3## 
where 
EQU e.sub.j= 0.01 Re(Vj.sub.meas)+0.03 
.vertline.Im(Vj.sub.meas).vertline.+0.004 
Here Vj.sub.homog is the voltage that would be recorded on channel j in an 
infinite homogeneous medium with a conductivity s.sub.f. The expression 
for ej is an estimate of the standard deviation in the measurement for 
channel j. In the expression for E(s.sub.f,x), the summation over j 
extends from 1 through 8, which corresponds to voltages recorded on the 
four receivers closest to the transmitter. By restricting the summation to 
the first eight channels, one obtains a better estimate of the formation 
conductivity close o the borehole. For the off-center AIT, the borehole 
signal is affected by the formation conductivity, and depends primarily on 
the formation conductivity immediately surrounding the borehole. 
Theoretically, if Vj.sub.meas =Vj.sub.model +Vj.sub.homog then E(s.sub.f,x) 
would be zero and the model voltages would match the measured values 
exactly. The value of E(s.sub.f,x) is stored in solver 40a4. 
The solver 40a4 selects a second set of values for the remaining two 
borehole parameters s.sub.f and r, yielding a second selected set of 
borehole parameters s.sub.m, s.sub.f, r, and x. The method used to select 
a new set of values for s.sub.f and r is the Levenberg-Marquard 
optimization method described in the following reference: D. W. Marquard, 
Journal of the Society of Industrial and Applied Mathematics, Volume 11, 
pages 431-441, 1963. The second selected set of borehole parameters is 
passed to the correction module 40a5, and the aforementioned process 
repeats itself once again. The correction module 40a5 generates a second 
model which comprises another set of eight complex numbers Vj.sub.model, 
j=1, . . . , 8, that are a function of the second selected set of borehole 
parameters s.sub.m, s.sub.f, r, and x, the values of VJ.sub.model being 
evaluated by the subroutines VDM06D, VDM72C as before. The eight complex 
numbers associated with the second model are transmitted backed to solver 
40a 4 and are compared, therein, with the eight complex numbers associated 
with V.sub.meas, using the least squares criterion described above. 
Another resultant value of E(s.sub.f,r) is stored in the solver 40a4. The 
aforementioned process repeats itself until a minimum value of 
E(s.sub.f,r) is found. When E(s.sub.f,r) begins to increase, the s.sub.f 
and r values associated the immediately preceding value of E(s.sub.f,r) 
are determined to be the desired values for the conductivity of the 
formation (s.sub.f) and the radius of the borehole (r). 
The objective of the above-referenced iterative process is to select values 
for s.sub.f and r such that V.sub.model most nearly matches V.sub.meas. 
The desired values of s.sub.f and r are passed directly to the final 
correction module along with the given values of s.sub.m and x, thereby 
making a final and complete set of borehole parameters s.sub.m, s.sub.f, 
r, x. A model voltage (Vj.sub.model) is evaluated for each channel j by 
the final correction module 40a7, as a function of the final set of 
borehole parameters: Vj.sub.model (s.sub.m,s.sub.f,r,x). The correction 
module 40a7 is identical with 40a5, except that it computes Vj.sub.model 
for j=1, . . . , 18, whereas 40a5 computes the first 8 values of j only. 
The final correction algorithm module 40a7 applies a borehole correction to 
the eighteen (18) chanels as follows: 
EQU Vj.sub.corr =Vj.sub.meas -Vj.sub.model (s.sub.m,s.sub.f,r,x) , j=1, . . . , 
18. 
Vj.sub.corr represents the "borehole corrected data", a set of 18 corrected 
complex voltages output from the final correction algorithm 40a6. 
In summary, Vj.sub.meas represents the complex voltage in the 18 receiver 
channel coils 18 and is contaminated by a contribution from the bore hole 
fluid ; whereas Vj.sub.model (s.sub.m,s.sub.f,r,x) represents the value 
predicted by the model for the contribution of the conductive borehole 
fluid. Vj.sub.corr represents the complex voltage in the 18 channels 
WITHOUT the effect of the conductive borehole fluid. The AIT data are 
therefore automatically corrected to represent the true formation 
conductivity and are not contaminated by the conductivity of the mud in 
the borehole. 
The AIT borehole correction software (hereinafter called the "BORCOR" 
program) was tested by performing a log in a well with an irregular 
borehole in a low-conductivity chalk formation. The results are shown in 
FIGS. 7 though 10. FIG. 7 shows the the raw signals recorded by the AIT in 
a 400-foot section of the well where the formation conductivity is known 
to be less than 25 mS/m. The mud conductivity in this section of the well 
is about 14000 mS/m. FIG. 8 shows the AIT signals after a simple borehole 
correction assuming a constant formation conductivity, and using a 
borehole radius as determined by a mechanical caliper. The borehole signal 
has been greatly reduced, but there are residual errors of the order of up 
to 500 mS/m. FIG. 9 shows the AIT signals after a borehole correction 
where the borehole radius and the formation conductivity are 
simultaneously optimized by the nonlinear least squares procedure in the 
BORCOR program. The residual errors have now been reduced to less than 100 
mS/m, or than 2 percent of the raw signal in FIG. 7. The borehole radius 
determined by the optimization in BORCOR is compared in FIG. 10 to the 
diameter recorded by the mechanical caliper. One reason that the least 
squares procedure gives better results can be seen in FIG. 10; the 
mechanical caliper has limited resolution, its accuracy is limited by 
inertia, and its readings have a coarse quantization. The optimized 
borehole corrections can better cope with a borehole whose cross-section 
is elliptical or irregular. 
APPENDIX A 
The purpose of this appendix is to describe fully the organization of the 
borehole correction software and explain the usage and purpose of each 
subroutine and each variable within those subroutines. The document is 
divided into sections called Prologues which describe each subprogram of 
the software. Each Prologue contains a general description of the routine, 
and an explanation of all of the variables and arrays used. 
Following the prologues is a source listing for both versions of the 
software, centered and decentered, including the driver routines. 
The driver routine must be written to accomodate the format of the input 
logs. The driver is very simple to write; only the routine BORCOR needs to 
be called, all other routines described in this document are called 
internally by BORCOR. Common blocks are all internal to BORCOR. 
##SPC1## 
The invention being thus described, it will be obvious that the same may be 
varied in many ways. Such variations are not to be regarded as a departure 
from the spirit and scope of the invention, and all such modifications as 
would be obvious to one skilled in the art are intended to be included 
within the scope of the following claims.