Borehole Correction System for an Array Induction Well-Logging Apparatus

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.

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

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.

A full understanding of the present invention will be obtained from the detailed description of the preferred embodiment presented hereinbelow, and the accompanying drawings, which are given by way of illustration only and are not intended to by limitative of the present invention, and wherein:

FIG. 1 illustrates a wellbore in which the new array induction tool (AIT) is disposed, the AIT tool being connected to a computer in a logging truck wherein the borehole correction software the present invention is stored;

FIG. 2 illustrates a more detailed construction of the AIT tool of FIG. 1;

FIG. 3 illustrates a more detailed construction of the well-site computer having a memory wherein the borehole correction software of the present invention is stored, a central processor to perform the required arithmetic operations, and storage media to save the processed data;

FIG. 4 illustrates a more detailed construction of the borehole correction software of the present invention;

FIG. 5 illustrates the borehole parameters used by the borehole correction software;

FIG. 6 illustrates the various subroutines which comprise the borehole correction software of the present invention;

FIG. 7 illustrates raw signals recorded by an AIT tool in a borehole of an oil well where the formation conductivity is less than 25 ms/m;

FIG. 8 illustrates the AIT raw data signals after a simple borehole correction;

FIG. 9 illustrates the AIT raw data signals after an optimized borehole correction; and

FIG. 10 illustrates a set of curves representing borehole diameter versus sonde position.

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:

The functions c.sub.1, . . . , c.sub.8 have the following form:

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

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:

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.

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.