Note : Les descriptions sont présentées dans la langue officielle dans laquelle elles ont été soumises.
CA 02483611 2005-12-08
HIGH RESOLUTION ARRAY INDUCTION TOOL
This application is a division of Patent Application No. 2,355,083 filed
December 14, 1999 for High Resolution Array Induction Tool.
STATEMENT REGARDING FEDERALLY SPONSORED
RESEARCH OR DEVELOPMENT
Not applicable.
TECHNICAL FIELD OF THE INVENTION
The present invention relates to tools for electromagnetic induction well
logging instruments. More specifically, the present invention relates to
methods for
obtaining resistivity measurements at greater depths and with better vertical
resolution that has heretofore been possible. Still more particularly, the
present
invention relates to a logging tool and operating system therefor that
provides high
resolution resistivity measurements using a multi-receiver array and novel
data
processing techniques.
BACKGROUND OF THE INVENTION
In petroleum drilling, it is often desirable to survey the formation using a
logging tool lowered through the wellbore. Electromagnetic induction well
logging
instruments are used to make measurements of the electrical resistivity of
earth
formations penetrated by wellbores. Induction well logging instruments
typically
include a sonde having a transmitter coil and one or more receiver coils at
axially
spaced apart locations from the transmitter coil.
The basic element in all multi-coil induction tools is the two-coil sonde. The
two-coil sonde consists of a single transmitter coil and a single receiver
coil
wrapped around an insulating mandrel. The transmitter coil is driven by an
oscillating current at a frequency of a few tens of kilohertz. The resulting
magnetic
field induces eddy currents in the formation which are coaxial with the tool.
These
eddy currents produce a magnetic field which in turn induces a voltage in the
receiver coil. This induced voltage is then amplified, and the component of
the
voltage that is ih-phase with the transmitter current is measured and
multiplied by
a tool constant to yield an apparent conductivity signal. This apparent
conductivity
is then recorded at the surface as a function of the depth of the tool.
The two-coil sonde has several practical limitations. Its response is
adversely affected by several factors including the borehole, adjacent beds,
and
mud filtrate invasion. Also, the two-coil sonde is difficult to implement
because of
the large direct mutual coupling between the coils. Even though this mutual
signal
is out of-phase with the transmitter current, it is a problem because a very
small
phase shift in the electronics can cause this mutual coupled signal to "leak"
into
the apparent conductivity signal. For these reasons, it is the standard
practice in
the industry to construct induction logging tools with coil arrays which
include
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CA 02483611 2005-12-08
additional coils. Typically, there are several transmitter coils and several
receiver coils. In
certain applications, all of the transmitter coils may be connected. in series
into one circuit.
Similarly, all of the receiver coils may be connected in series in a separate
circuit. The
additional coils served to cancel out the various adverse effects listed
above. Such arrays are
generally termed "focused arrays."
The following are terms of art that are used often to compare various
induction tools.
The "vertical resolution" of a tool is a measure of the thinnest bed that a
tool can detect. That
is, a tool may accurately estimate the thickness of beds that are thicker than
its vertical
resolution. A tool can also accurately locate a bed boundary to within the
tolerance of its
vertical resolution. There is still a significant error in the apparent
conductivity reading in a
thin bed, which is attributable to signals from adjacent beds; however, so
long as the thin bed
is thicker than the vertical resolution of the tool, the tool can estimate the
thickness of the
bed. The error in the apparent conductivity reading of a thin bed attributable
to signal from
adjacent beds is referred to as "shoulder effect." In known induction tool
arrays, the
additional coils are arranged to cancel out much of this shoulder effect.
It is also possible for a tool to have good vertical resolution but poor
shoulder effect.
Such a tool would be able to accurately define bed boundaries but would give
poor estimates
of the conductivities of these thin beds. Vertical resolution and shoulder
effect are two aspects
of the vertical focusing of an induction tool coil array.
The "depth of investigation" of a tool is a measure of the average radius of
penetration
of the signal. The "depth of investigation" is defined as the radius of the
cylinder from which
half the apparent conductivity signal comes. The "borehole effect' is a
measure of hover much
signal comes from the borehole as compared to the formation. In conventional
arrays, coils .
are arranged to cancel much of the signal coming from near the tool so that
the "depth of
investigation" will be large and the "borehole effect" will be small.
The foregoing discussion is based on an assumption that a tool can be operated
at a
sufficiently low frequency that there is no significant attenuation of the
transmitted signal as
the signal propagates ~ through the formation. In practice, such attenuation
cannot be
neglected, since it reduces the transmitted signal proportionately more in
conductive
formations. The voltage actually induced in the receiver coils is typically
less than what
would be induced for any value of conductivity were the relationship between
eddy current
magnitude and the induced voltage a linear one. The difference between the
voltage actually
induced and the voltage that would have been induced if the relationship were
linear results
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CA 02483611 2005-12-08
t
from the so-called "skin effect." Prior art practitioners generally attempt to
design a coil array
which has moderate skin efl'ect at the highest conductivity of interest in
logging situations and
then correct for the skin effect at the surface. The skin effect correction is
typically a
correction which yields the true conductivity of a homogeneous formation.
In the case of conventional induction tool arrays, coils must be positioned to
define
the tool's vertical resolution, depth of investigation, as well as to
compensate for borehole and
shoulder effect. In addition, the coils must minimize the mutual coupling
between transmitter
coils and receiver coils, as this signal is very large when compared to most
formation signals.
In known coil arrays, the position and strength of each coil controls each of
these
IO aforementioned effects. Because each of these effects may change as a coil
is modified, it is
difficult to design a coil array optimized to reduce all of these effects
simultaneously. The
different effects interact, as one effect is reduced, another is increased.
Conventional coil
array designs therefore must be a compromise between sharp vertical resolution
and deep
radial penetration into the formation. In addition, in some prior art tools,
the deep
measurements lack sufficient vertical resolution, so the high resolution
shallow measurements
are used to enhance the resolution of the deep coil measurements: This is
undesirable,
however, particularly when the shallow measurements become corrupted. It would
be
desirable, therefore, to provide an induction logging tool that permits both
sharp vertical
resolution and deep radial penetration.
In most commercial applications, it is also desired to investigate the strata
surrounding
a borehole to different depths, in order to determine the diameter of invasion
of the strata by
borehole fluids. This requires at least two measurements with contrasting
radial response and
ideally identical vertical resolution so that differences in the logs obtained
will be due to .
radial anomalies in the formations, such as invasion. Mast prior art dual
induction tools use
deconvolution filters to match dissimilar vertical responses of two induction
coil arrays with
inherently different vertical resolutions by smoothing out the response of the
array with the
sharper vertical resolution and degrading it to match the vertical response of
the second array.
This approach is not desirable in view of the degradation of vertical
resolution that is needed
to match the different coil arrays.
In another prior art system, separate deep transmitter coils and medium-deep
transmitter
coils are provided. Because the cross-talk would obscure any signal received
from either set of
transmitter coils, it is necessarc~ to use a time multiplexing approach. In
the time-multiplexing
approach, the sets of transmitters are turned on alternately and a settling
period is allowed
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between signals, so that only one signal path is in use at any time. The time-
multiplexing
approach becomes less practical, however, as the number of coils increases,
and is impractical
when six or more sets of transmitter coils are used.
Other prior-art systems for array induction logging have a single transmitter
coil and sets
of receiver coils above and/or below the transmitter. Each set of receiver
coils consists of a main
receiver coil and a bucking receiver coils. The main coils are positioned at
different distances
from the transmitter and the coil sets are arranged such that the distances
between the main and
bucking receivers increase with increasing distance from the transmitter. In
such a system, the
vertical resolution for the deeper arrays is inferior to the vertical
resolution of the shallower
arrays. With such a system, the measurements from the deep arrays must be
combined with
information from the shallow arrays in order to produce a deep. response with
good vertical
resolution. The shallow arrays are more affected by temperature and the
borehole, and this
system causes errors in the data generated by the shallow arrays to degrade
the accuracy of the
deep logs. '
Thus, a need exists for an array induction tool capable of investigating
multiple depths
of investigation while maintaining substantially identically vertical
resolution for all coil
arrays. It is further desirable to provide a tool capable of separating the
vertical and radial
aspects of the signal processing. A more detailed discussion of these and
related problems
can be found in U.S. Patent 5,065,099;
BRIEF SUMMARY OF THE INVENTION
The present invention includes a coil array and signal processing system that
pennit
sharp vertical resolution and deep radial penetration. Further, the present
system is capable of
investigating multiple depths of investigation while maintaining substantially
identically
vertical resolution for all coil arrays. The present coil array and signal
processing system
allow 10, 20, 30, 60, 90 and 120 inch depths of investigation, with of one and
two foot
vertical resolutions.
' The present tool includes a plurality of coils spaced along the tool body at
preferred
intervals. Several of the coils arc shared between coil sets and several of
the coils may be
tapped, so that, for example, ten elemental measurements can be made by 19
coils. The size,
spacing and direction of winding of the coils allows the present signal
processing system to
calculate a weighting system that yields conductivity measurements for the
preferred depths
of investigation. The preferred signal processing system digitized the
received waveforms
and extracts phase information from the digitized signal.
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According to a preferred embodiment, each elemental measurement is deconvolved
vertically to match resolution before any radial combination occurs. Further
according to the
present invention, the output deep measurements (90 and 120 inch depths of
investigation) are
constructed from only the deepest of the elemental measurements and are not
corrupted with the
shallow elemental measurements.
BRIEF DESCRIPTION OF THE DRAWINGS
For a more detailed understanding of the preferred embodiments, reference will
now be made
to the Figures, wherein:
Figure I is a schematic diagram of a prior art induction array;
Figure 2 is a schematic diagram of an induction array constructed in
accordance with
the present invention;
Figure 3 is a wiring diagram for the upper portion of the array of Figure 2;
Figure 4 is a plot of vertical geometric factors for the array of Figure 2;
Figure 5 is a plot of integrated radial geometric factors for the array of
Figure 2;
Figure 6 is a plot of integrated radial geometric factors calculated using
selected
weights for the array of Figure 2;
Figure 7 is a plot of the integrated radial geometric factors of Figure 6,
shown in
exponential scale;
Figure 8 is a series of plots illustrating one embodiment of the present
digital data
extraction;
Figure 9 is a portion of a theoretical waveform and representative A/D samples
generated using the present system;
Figure 10 is a plot of the results of the present data sampling system;
Figure 11 is a plot showing the averaged data from Figure 9 compared to the
theoretical waveform;
Figure I2 illustrates how log responses can be assembled in matrix form to f
nd the
unknown filter coefficients; and
Figure 13 is a plot showing results.of a hypothetical system deconvolved with
the new
algorithm.
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DETAILED DESCRIPTION OF THE INVENTION
Refernng initially to Figure 1, one type of known coil array uses a single
transmitter
and two trios of receivers 11, 12 (one trio above the transmitter and one trio
below the
transmitter), which are connected to and measured by two separate channels in
the signal
5 processor. Each trio of receivers 11, 12 includes a main receiver and two
bucking receivers
that are equally spaced on either side of the main receiver. A detailed
discussion of three-
receiver sets can be found in U.S. Patent No. 5,065,099. 'The bucking
receivers have equal
turns. The signals from the two trios are measured separately and depth-
shifted to align with
the center receiver and then averaged. In each trio of receiver coils, the
outer bucking
10 receivers are typically wound in series opposition with the central main
receiver. The upper
and lower trios of receivers 11, 12 are wired separately and measured by
separate electronics
channels. Thus, the single, central transmitter can be energized continuously.
The advantage
to this approach is that, because it uses a single transmitter, there can be
additional trios of
receivers at different spacings with all receiver trios measured
simultaneously.
i 5 An array induction log is composed of a number of raw elemental
measurements.
Each elemental measurement is a simple induction tool. The spacings of the
elements must
be chosen to cover a wide range of depths of investigation with sufficient
coverage to be
combinable to produce the desired 10, 20, 30, 60, 90 and 120 inch depths of
investigation.
The configuration of coils and the resulting measurements so as to achieve a
target depth of
investigation with minimal borehole effect are discussed in detail below.
In preferred embodiments of the present invention, the number of receiver
trios is
increased to correspond to the number of measurement depths, if desired, and
the coils can be
configured to provide optimal signal combinations for the desired measurement
radii. In
addition, it has been discovered that, although a two-receiver measurement is
deeper than a
three-receiver measurement for a given main transmitter-receiver spacing, a
three-receiver
measurement has better vertical resolution than a two-receiver measurement.
Radial Profiline
According to the present invention, the weighted average of several elemental
coil
measurements is used to achieve a radial response that is better than any
single elemental
measurement. Although the actual array induction processing is two-dimensional
(it includes
vertical as well as radial deconvolution) in thick beds, the logs from each
elemental
measurement are constant, and the effects of the vertical and radial
deconvolution can be
considered separately.
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As is known to those skilled in the art, Doll's integrated radial geometrical
factor (IRGF)
provides a simple induction tool model for cylindrical geometry that is valid
in the limit of low
formation conductivity or low tool operating frequency. Doll's IRGF is zero at
zero radius and
increases to one at infinite radius. It is a measure of how much signal comes
from inside
S cylinders of varying radii. The apparent conductivity response Qe to a step
invasion profile at a
radius r is given by
Qa(r) ° G(r) Q'xo 'f' ( 1 - G(r)~ at ( 1 )
where a~ and QX° are the conductivities of the uninvaded and invaded
zones, respectively, and
G(r) is the integrated radial geometric factor (IRGF) at radius r. If there
are n elemental coil
measurements, the IRGFs can be combined linearly to produce a composite
response given by
n
G~ (r) _ ~ w;G, (r) (2)
,n
where
n
~wf -1 (3)
r=i
The weights must sum to unity for the composite measurement to read correctly
in a
uniform formation. Given a set of elemental coil responses there must be a set
of iv; that best
achieve a given target response G~(r). According to the present invention, a
target composite
response G~(r) is defined, and then the system is solved for the set of
weights w; that best
achieves this target response. The apparent conductivities (the elemental
measurements can
be combined in the same way as the Doll IRGFs.
For purposes of analysis, an ideal IRGF was assumed to be a step function that
goes
from zero to one at some radius. To achieve this response, all of the signal
must be confined
to a thin cylindrical shell. This ideal response cannot be achieved because
deeper coil
measurements are less resolved radially than are shallow ones. Thus, the best
compromise is
to set a target radius point where the IRGF should reach the value 0.5. The
IRGF should stay
near zero inside a maximum borehole radius, pass through the target 50% point
and then go
to one as soon as possible. The shape of the curve between the maximum
borehole radius and
the radius where the 1RGF reaches one is relatively unimportant so long as the
IRGF is
monotonic and stays between zero and one. Past this radius, the IRGF should
remain near
one.
A nonlinear least-squares function minimizes is one preferred method for to
determine
the unknown w;, although other suitable methods will be known to those in the
art. The
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function minimizer iteratively varies the unknowns to minimize the sum of the
squares of an
array of user supplied functions. Because there are more functions than
unknowns, the
system is over-determined. Also, although this is a linear problem, the
nonlinear minimi2er
allows us to add penalty functions to discourage certain solutions.
S According to one preferred embodiment, the error function to be minimized is
derived
from the IRGF factor itself. A function y(r;) that can be evaluated at
different radii r; to
produce an array of functions y; to be minimized by varying the unknown
weights is defined
as follows:
lOBG,, for r < r,,
G(r) for r,, < r < r~~
y(r) ~1 for r = rso,, (4)
_ /
2
-
G(r)~
[1 foY r5pn/e < r < 3r5%
'
G(r)1
1061- G(r)~for r >_ 3r5~
where the radius rH is the desired maximum borehole radius. The radius rs~,o
is the target 50%
radius, the radius at which G reaches one-half of its total value. The factors
of 108 and 106
were chosen empirically. The radius 3r5o~,o is the smallest radius of a
cylinder that contains
nearly all the signal.
In a preferred embodiment, additional logic is used to penalize geometrical
factors
less than zero or greater than one, so as to discourage undershoots or
overshoots. The error in
the 50% point rso~~ is appended to the vector of functions to be minimized. In
addition, the
noise gain,
~~z
~w,z
r=i
is appended to the vector of functions to be minimized. This favors solutions
with weights of
low magnitude to minimize the noise gain, as solutions with large positive or
negative
weights or highly oscillatory weights are undesirable. The weighting factor
for these
additional measurements can be chosen by trial and error. There are n-1
unknowns, since the
weights must sum to unity. The radii can be fixed so the IRGF of each set can
be tabulated
before any optimization.
The MINPACK routine LMDIF1, available from Argonne National Laboratory was
found to work well for this application. The LMDIFl routine minimizes the sum
of the
squares of m nonlinear functions in n variables by a modification of the
Levenberg-Marquardt
algorithm. This is done by using the more general least-squares solver LMDIF.
The user
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CA 02483611 2005-12-08
must provide a subroutine that calculates the functions. The Jacobian is then
calculated by a
forward-difference approximation. In the present application, the minimization
program
reads in an arbitrary number of coil sets from disk files, tabulates their
IRGF, inputs the
desired 50% point and maximum borehole diameter. The output is the set of
weights and
plots of the composite IRGF on regular (invaded zone) and expanded scales
(borehole
region). It will be understood that other suitable routines can be substituted
for LMDIF1.
It has been discovered that the six depths of investigation that are desired
in the
preferred embodiment, namely 10, 20, 30, 60, 90 and 120 inches, can be
effectively measured
using the output from four elemental measurements. Although only four
elemental
measurements are required, additional measurements will reduce the noise of
the achieved
actual achieved responses. It is desirable to make the deep measurements as
sharp as possible,
so that the deep measurements stand alone and do not require the shallow
measurements to
enhance vertical resolution. The nonlinear solver can be run on each of the
sets to create depths
of investigations of 10, 20, 30, 60, and 90 inches with minimal borehole
effect and minimal
noise gain.
In general, a two-receiver measurement is deeper than the corresponding three-
receiver measurement by about 20%. Similarly, a logarithmic spacing of the
receivers covers
the range of depths of investigation more economically, i.e., it concentrates
the measurements
at shallow spacings, where the response varies rapidly. There are many
possible combinations
of weights that will achieve equivalent shallow responses.
There is little to be gained from any spacing whose depth of investigation is
greater than
the deepest synthesized depth of investigation. For example, in one three-
receiver set, a main
spacing of about 69 inches produced a 90 inch radius of investigation, with no
need of radial
combination that risks corruption by shallower measurements. Likewise,
shallower spacings are
more sensitive to variations in tool position in the borehole. Therefore, it
is desirable to make
the shallowest measurement as deep as possible while still being able to
achieve the smallest
desired depth of investigation. This is because a shallow response achieved by
subtracting two
deeper responses will have less borehole effect.
As stated above, two-xeceiver measurement is deeper than a three-receiver
measurement for a given main transmitter-receiver spacing. This is not a
limitation so long as
there is one elemental measurement with a depth of investigation greater than
90 inches. The
reduced depth of investigation of the three-receiver measurement is actually
an advantage for
the shallow depths of investigation, since it allows them to read shallower
with the same
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CA 02483611 2005-12-08
spacing. For the intermediate spacings, there is not enough difference in the
radial response to .
prefer one over the other. In order to ensure that the deep measurements are
inherently sharp,
the three-receiver system is preferred. Within this system, the logarithmic
spacing provides
the best coverage.
With the present system and method, it is possible to combine the responses of
deep
and shallow measurements to achieve delicate cancellation in the region of the
borehole.
There are some real-world considerations that make this undesirable, however.
Doll's model
assumes a tool centered in a round borehole, for example. The standoff is not
measured
directly, nor is the shape of the borehole. In addition, the uncertainty and
variation in tool
position and borehole shape contribute noise to the measurements, which
contributes to the
residual uncorrectable borehole effect. This noise is much more of a problem
with the
shallow coil measurements than with deep measurements. If the signal
processing for a
synthesized output curve contained a mixture of deep and shallow elemental
coil
measurements, the composite could be corrupted by the uncorrectable borehole
effect of the
1 S shallow measurements even if the composite output has a deep response. For
this reason, it is
preferred to allow a small borehole effect on the composite if it allows one
to avoid using the
shallow measurements. Hence, the deep output curves are preferably composed of
only the
deep elemental measurements.
Preferred Array
Based on the foregoing preferences, a set of elemental measurements (coil
array) can
be developed. One preferred set of ten elemental measurements is shown in
Figure 2. Each
shaded rectangular block represents a coil. Tapped coils are indicated with a
line drawn
through the shaded area of the coil. The main receiver coil of each elemental
set is numbered
and each main receiver is bracketed with its bucking receiver{s). As shown in
Figure 2, a
preferred array includes ten elemental measurements, namely 110, 120, 130,
I40, 150, 160,
170, 180, 190, and 200. In Figure 2, the main receiver coils are numbered 1-6.
The deepest three-receiver set 120 preferably has a 72 inch spacing to main
coil 102
and a 9 inch spacing to bucking receivers .104, 106. The end bucking coil 104
for the three-
receiver set becomes the main receiver for the two-receiver set 120. A tapped
portion of the
main receiver 102 of the three-receiver set can be used as the bucking
receiver for the deepest,
two-receiver set 110. The preferred two-receiver measurement has a depth of
investigation of
about 120 inches.
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The next shallowest array 130 can share a bucking receiver 104 with the
deepest
three-receiver set 120. It is d$sirable to improve the vertical resolution of
this measurement
by decreasing the bucking receiver spacing to six inches, since sensitivity is
less of a problem
with shallow spacings. This makes approximately the same sensitivity for the
last three
arrays 140, 150, 160. The remainder of the elemental measurements continue
with the six
inch bucking receiver spacing and share all or a portion of the: bucking
receivers of their
nearest neighbors. The shallow arrays have a surplus of signal. In order to
save wire, it is
desirable to use fewer turns on the shallower arrays. For this reason, an end
tap is placed on
one of the bucking receivers.
The lower receivers 170, 180, 190, 200 are preferably minors of upper
receivers 110,
120, 130 and 140, respectively. The two shallowest upper receiver arrays 150,
160 are not
mirrored in the lower coils, although in an alternative embodiment, the array
could be made
fully symmetric.
Referring now to Figure 3, each electronics channel is preferably connected to
a pair
of wires. For example, the receiver electronics for receiver channel 3 are
connected to the
two wires labeled "3" at the top. One advantage of the preferred arrangement
of shared coils
is that it has the potential to reduce the number of connections that feed
through the bulkhead.
By suitably arranging the taps and common connections, the upper receiver
coils 110, 120,
130, 140, 150, 160 can be connected with only seven wires, as shown-in Figure
3, in which
the main receiver coils again numbered 1-6. The wiring bundle is preferably
twisted together
between each coil and under each coil:
Specific details of an alternative exemplary configuration for the tool array
are set out
below. Note that coils with minus sense are wound in the opposite direction
than the
remainder of the coils. This embodiment differs from the one shown in Figure 2
in that the
30 inch receiver set has been moved from above the transmitter to below the
transmitter, with
the result that the 18 inch set. which remains above the transmitter, does not
share any coils.
For an array having an effective coil diameter of 2.695 inches, a 0.066 inch
winding pitch,
and an 1.100 inch diameter feed pipe Table 1 gives the preferred numbers of
turns and
preferred positions of each coil center (based on filamentary loop coils, one
loop per turn).
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Table 1
Coil Set Turns Inches
Transmitter 200 00.000
Receiver Array i at 47 78.035"
78"
(above and below transmitter)-29 66.525"
**
Receiver Array 2 at -47 78.03
69" 5"
(above and below transmitter)104 69.000"
-47 59.950"
Receiver Array 3 at -47 59.950"
54"
(above and below transmitter)1 O1 54.000"
-47 47.988"
Receiver Array 4 at -47 47.988"
42"
(above and below transmitter)106 42.000"
-47 35.968"
Receiver Array 5 at -11 34.780"
30" **
(below transmitter only}29 30.000"
-I 24.135"
1
Receiver Array 6 at -I 24.135"
18" 1
(above transmitter only)25 18.000"
~
-11 15.056"
*shared coils
**end-tapped coils
Raw vertical and integrated radial geometric factor (VGF and IRGF) plots in
Figures 4
and 5 show the response of the foregoing elemental measurements. These were
computed
based on point receiver coils but with the single transmitter coil split into
several point
dipoles. The weights calculated to achieve the target radial responses are
shown in Table 2.
Blank entries mean that that elemental measurement is not used. Again,
elemental array 1 is
deepest and 6 is shallowest. These weights achieve the IRGF responses shown in
Figures 6
and 7. It will be understood that the numbers, size, spacing and weights for
the coils in the
array can be varied, and can be optimized for various desired target depths.
Table 2
rso~io wr ~'a ~'3 wa ws
120 1.126 -0.126 - - - -
90 - 1.045 -0.045 - - -
60 - -0.569 1.569 - - -
30 - -1.567 1.075 0.865 0.627 -
- -0.25 -0.569 -0.036 1.506 0.348
10 - - 0.380 -0.029 -1.492 2.142
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Extractin P~Information from the Di aitized Waveform
In order to simplify the electronics needed to process the received signal, it
is
preferred to digitize the receiver waveforms and to use digital signal
processing to perform
the phase-sensitive detection. This eliminates the heavy analog filtering
required by the
analog phase-sensitive detectors, which forms a substantial portion of the
electronics in prior
art devices. Hence, a preferred embodiment of the present system includes a
digital signal
processor (DSP) that performs filtering and phase-detection downhole.
In order to obtain meaningful information from the received signals, it is
necessary to
make phase-sensitive measurements of the receiver voltages. This entails
measuring the
portion of the receiver voltage that is in-phase with the transmitter current.
One preferred
method is to measure both the transmitter current and receiver voltage with
respect to an
arbitrary phase reference and then divide the two complex voltages. To do
this, it is necessary
to measure in-phase and quadrature voltages with respect to the arbitrary
phase reference.
Fast A/D converters and microcontrollers or DSPs are now available to digitize
the
receiver waveform and to do the phase-sensitive detection digitally. , Some
prior art devices
digitize the waveform from each receiver. These devices stack waveforms and
use an FFT to
extract the components at the different frequencies. In contrast, the
preferred embodiment of the
present tool uses only two frequencies. A novel, preferred method of stacking
the waveforms
and extracting the R and X signals at two frequencies from the digitized
waveform is described
in detail as follows.
The preferred technique stacks a large number of waveforms to produce an
averaged
waveform. This averaged waveform can be plotted for diagnostic purposes if
desired, but the
primary goal is to measure the two phase components of the two frequencies of
interest. The
period of this averaged waveform is the period of the lowest frequency of
interest, f ,.
According to the present technique, the waveform must be sampled at a rate
that is at least
twice the highest frequency of interest, fH. The highest frequency must be an
integer multiple
of the lowest frequency for the composite waveform to be periodic with the
period of the
lowest frequency. The strategy is to divide the period of the lowest frequency
into B bins.
Each bin represents a discrete phase and its purpose is to average together
all A/D samples of
like phase that are sampled over a certain time window. Each bin is 360/B
degrees apart in
phase of the low frequency fL. The waveform is sampled every B' bins, where B'
and B are
relatively prime, i.e. have no common factors. The value of B' is referred to
herein as the
13
CA 02483611 2005-12-08
"stride." The time difference between two bins is 1/(B ~ fL). The A/D sampling
rate is (B
fi~)B'. To satisfy the Nyquist criterion, this rate must be more than twice
the highest
frequency, so (B ~ fL)B' > 2 fH. If B' and B have no common factors, then
after B samples,
the sample number (B+I ) will be placed in the same bin as the first sample.
If B' and B are
chosen in error to have a common factor; this wrap will happen sooner, and
bins will be
skipped.
In operation, the stacking process proceeds as follows. A digital waveform
generator
generates the frequencies f~ and f .,. There could be a separate generator for
each frequency, or
the sum of the two waveforms could be stored in the generator. There could be
other
. frequencies between f~ and fH, but they must be integer multiples of fL. The
waveform
generator output is amplified and sent to the transmitter coil(s). There is
preferably a separate
amplifier and A/D converter for each receiver coil, as well as a separate A/D
that digitizes the
transmitter current. The AID and the D/A converters are preferably clocked
with a single
clock; so that the measurement system is locked in phase with the transmitted
waveform. The
downhole processor keeps track of the current bin. For each channel, the
current AID sample
is added to the contents of the current bin and the count for that bin is
incremented. Then the
bin number is incremented by B'. If the result is greater than B, then B is
subtracted. This
process repeats. When an answer is desired, an average for each bin is
computed, and the bins
are initialized to zero.
As a result of the foregoing process, A/D samples at the same phase are summed
and
averaged together in each summing bin. The resulting average represents a
waveform that is
sampled more finely than actual sampling rate and can be plotted for
diagnostic purposes, if
desired. To extract the R component of the frequency f, which is either fL or
fH or a multiple
of fL between fL and fH, the averaged waveform x; for the i'h bin is
multiplied by the sine of the
angle (2xf i / B) where, f is the frequency of interest, and the product is
summed.
a-i
VR = ~ sin(2nf t )x, (6)
=o
s_~ (7)
Yr = ~ cos(2nJ' ~ )x;
i=o
By way of example only, the sine and cosine coefficients can be stored in a
look-up
table. This amounts to taking the dot product of the bin average vector and a
"filter" vector
containing the proper trigonometric function.
14
CA 02483611 2005-12-08
At the conclusion of each measurement cycle, VR and Va~ for each frequency and
channel are transmitted to the surface. The values of V~ and YX for the two
frequencies can be
used to derive the phase and magnitude of the two frequency signals:
Specifically, the phase
is given by tari ~(VX l 1~,~) and the magnitude is given by (YR 2+ YX Z)~~.
The bins are then cleared, and the process is repeated. During the time the
summation
and data transmission are performed, as during data accumulation, the
transmitted waveforms
continue and the bin number is incremented, so as to keep the bin count in
sync with the
transmitted waveform.
Figure 8 illustrates the foregoing process on a pair of hypothetical
waveforms. Plots
(A) and (B) are plots of pure signals at 8.0128205 kHz and 32.051282 kHz,
respectively, such
as might be used to drive a transmitter. The actual waveform seen by each
receiver will be a
sum of these two waveforms with arbitrary phase shifts on either one. The
actual received
waveform will also contain random thermal noise as well as other noise. The
dots on each
plot represent A/D sampling. Because Figure 8 illustrates an embodiment in
which 39 bins
are used (B = 39), in each waveform (A) and (B), the sample number 0 has the
same value
and phase as sample number 39.
After a predetermined number of cycles through the shuffling process shown in
8(C),
the contents of each bin are averaged, (i. e. the contents are summed and the
total is divided by
the predetermined number of cycles) and the results are one cycle of the ~8
kHz waveform
(E) and 4 cycles of the ~32 kHz waveform (F). After the shuffling process, the
waveforms
are more finely sampled in phase by a factor of 4 and the data are effectively
compressed into
one-fourth (1/B') as many cycles as were encompassed by the original number of
bins.
After averaging the contents of each bin, most of the noise averages out to
nothing.
The thermal noise and quantization errors are averaged out in two stages. If,
for example, (fH
- i) = 32 kHz, it is necessary to sample at a rate of at least 64 kHz to
satisfy the Nyquist
criterion. For a 50 ms acquisition time, this corresponds to at least 3200 A/D
samples. If
there are 100 bins, so that B = 100, each bin will receive 32 samples; for a 6-
fold reduction of
noise. The dot product further reduces the noise by a factor of Bra.
Example
This phase separation process can be modeled. One exemplary model uses
frequencies
of 8 and 32 kHz. The following simulations were performed with B = 128 and B'=
7, giving a
sampling rate of 146.286 kl~z. The following figures assume a conversion time
of 0.050
seconds, 5 mmho/m per lsb, and 10 mmhos/m noise level. For R8 k,~Z = 10
mmho/m, xR kH, _
CA 02483611 2005-12-08
20 mmho/m, R32 ~;Hz =30 mmho/m, and X32 kHz = 40 mmho/m, Figure 9 shows a
portion of the
theoretical received waveform {the curve) and representative A/D samples (the
points). After
7296 samples, the acquisition stops, and there are 57 values in each bin.
Figure 10 shows the
contents of the bins. Several points overlay because of the discretization of
the A/D converter.
In Figure I1, the averaged contents of each bin (the points) are compared to
the theoretical
waveform (the solid curve): The measured waveform (reconstructed by
multipiying the -
measured voltages by the sines and cosines) is the dashed curve. Importantly,
the dashed
curve nearly overlays the theoretical curve. One can clearly see the two
levels of noise
reduction. The first is evident in that there is much less scatter in the
points on Figure 11 than
in Figure 10. The second level of noise reduction is evident in the degree to
which the
reconstructed waveform matches the measured waveform.
It will be understood that the shuffling process described above, vyhich
allows
extraction of digital data, allows finer sampling and eliminates noise, can be
performed in a
variety of ways. For example, instead of shuffling the sampled values before
averaging, the
I5 average Values could be shuffled or the weighting factors themselves can be
shuffled.
Regardless of how it is carried out, the foregoing A/D conversion scheme has
numerous
advantages. The R and X signals at the frequencies of interest are computed
simultaneously;
the algorithm gives a single set of measured voltages (or apparent
conductivities) that
represent an average over the whole acquisition time; the waveform is
effectively sampled
more finely than the AID sampling rate; the bin summing process can be
continuously
computed on the fly without a need to store the samples in memory; and the
waveform can be
plotted for diagnostic purposes if desired. .
The bandwidth of the process was found to be equal to the reciprocal of the
acquisition time. The ripple outside the band can be controlled by using a
weighted average
with some windowing function instead of a straight average.
In one embodiment, it was found to be desirable to generate the transmitted
waveform
and drive the AlD converter with the clocks derived from the same master
clock. This locks the
phase of the transmitted waveform and the phase detection system. One
particular A/D device
outputs A/D samples at a rate of 78.125 kHz when clocked with a frequency of S
MHz. It was
found expedient to use a frequency of 10 MHz to drive the waveform generator.
Both of these
frequencies were obtained by dividing down a higher frequency system clock.
From the choice
of coil spacings and the expected conductivity range and the number of powers
of conductivity
used in equation 13, it 'was found expedient to use a frequency fi, ~8 kHz.
With these
16
' ' CA 02483611 2005-12-08
constraints, a search for the values of B and B' found the following
combinations that yield
frequencies near 8 kHz:
B B' f , kHz
39 4 8.0128205128
77 8 8.1168831169
79 8 7.9113924051
155 16 8.0645161290
157 16 7.9617834395
Of these, the simplest choice (as well as the closest to the desired 8 kHz) is
39 bins stride 4.
This is the configuration illustrated in Figure 8.
The multiplication by the sines and cosines can take place at the surface or
downhole
in a microcontroller or DSP. If the contents of the averaging bins is sent to
the surface and
recorded, any frequency that is a multiple of f~ that is less than the half
the AID rate can be
extracted provided these frequencies are in the present in the spectrum of the
transmitter
current waveform.
Vertical Deconvolutian
Induction tools measure apparent conductivities that suffer from a ~ variety
of
- 20 environmental effects. Chief among these are skin effect and shoulder (or
adjacent bed)
effect. The two are intertwined. The vertical response varies with the
conductivity level.
Various methods exist for correcting for these effects. An effective system
requires a
deconvolution over depth. Most prior art deconvolution techniques are not well
suited to the
asymmetrical elemental measurements of an array tool. The present invention
includes a new
method of deconvolving the elemental measurements to match the vertical
responses prioi to
any radial combination and.a method for finding the corresponding filters.
Induction tools measure the voltage of a set of receiver coils typically wound
in series.
The receivers are arranged so that the direct mutual coupling from the
transmitters is nulled.
In low conductivity homogeneous formations, the voltage induced in the
receiver coils is
proportional to the formation conductivity. The voltage multiplied by the
appropriate "tool
constant" will produce an apparent conductivity that is correct in the low-
conductivity limit,
but will be less than the actual conductivity in high-conductivity formations.
This
nonlinearity is commonly called "skin effect."
In a homogeneous medium a two-coil sonde will read:
a~a = a- ao~~ - a, ~~ - a2a'~ ... (8)
17
CA 02483611 2005-12-08
To make the apparent conductivity read correctly in a homogeneous medium, a
function is
applied to the reading.
~ =.f(~e) (9)
Since Qh > ~a over the range where the tool is commonly used, this procedure
is called
"boosting." Before the advent of the digital computer, this boosting function
was
implemented as an analog function former. On a computer, there are various
ways of
constructing or tabulating the boosting function f. Since Qa ~ ~ in low
conductivities, the
following functional form has been found to be convenient.
I 0 ~ - Qe .~ CO Q.a3/2 ,+ C ~ ~a5/2 + C2 ~.a7/2 ~. . . .
( 10)
This can be rearranged to a power series in ae that can be truncated to a
polynomial:
(fib - Via) / 6a3/2 = CO ~. CI ~~- C2d'-F. ...
(11)
The linear least-squares technique can then be used to fit a polynomial to the
data from a
computer model over a selected conductivity range.
In addition to skin effect, induction tools suffer from shoulder or adjacent
bed effect.
The measurements at a particular depth are influenced by formations a
significant distance
above and below the tool. If one uses the correct homogeneous boosting
function, one will
find that the induction response is nonlinear. That is, if one generates a
simulated log through
a formation and then generates a second log through a fonmation that has all
of the
resistivities scaled by a constant, the second simulated log will not be
merely the first log
scaled by the same constant. The shape of the log varies with resistivity
level. It is necessary
to correct for skin effect and shoulder effect simultaneously so as to
linearize the response.
Several methods can be used. It is known to use the X-signal (quadrature
signal) to
accomplish this, as disclosed and described in U.S. Patent No. 4,513,376,
Another known method. disclosed and described in U.S. Patent No. 5,145,167,
which
is incorporated herein in its entirety, uses the apparent conductivity raised
to the power 3/2.
The formula used is
W >(Z)=a~a'ofZ)~~,.1'rya(Z-dr~'~~ha312(Z)~~~'y'p3iz(Z_d~)
( 12)
18
CA 02483611 2005-12-08
The functions a and (3 work together to make the tool read correctly in a
homogeneous
formation. The deconvolutibn filters f and g work in concert to shoulder
effect correct the
tool. Finding the boosting functions a and ~i and the filter coefficients f
and g is a nonlinear
problem, since the boosting function multiplies the output of the convolution.
Because the
present tool is asymmetrical, prior art techniques for finding the filter
coefficients are
cumbersome. It is therefore desirable to provide an efficient method for
finding the filter ,
coefficients..
In a homogeneous formation, the functions a and ~3 are necessary to account
for the
higher order (greater than 3/2) terms neglected from equation (8). The
preferred method is to
use a different filter for each power needed in fitting equation (9). A
preferred formula is:
6v (Z)-~ ~fWpnl(Zrd~)'
j 1
(13)
where the powers p~ are given by p~ = 1, 3/2, 5/2, 7/2, 9/2, ...
The filter coefficients for a particular power should sum to the corresponding
homogeneous boosting coefficient above for the tool to read the correct value
in a
homogeneous medium.
~.~i ° c
a
( 14)
To find the unknown filter coefficients f j, one first fords a set of boosting
coefficients
c~ that approximate equation (2). One method is to use the least-squares fit
to the polynomial.
The more difficult problem is to ford the actual filter coefficients f,~. One
method of finding
the unknown coefficients f~ is to use several simulated logs at different
resistivity levels and
then use the method of linear least squares to find the coefficients that best
match a set of
target logs. A computer model that simulates an induction tool in a layered
formation can be
used to tabulate simulated logs. This amounts to solving the linear system of
equations Ax
b where the A matrix is filled with the simulated raw logs and the b vector is
filled with the
target log. The solution which minimizes the residual Ax - b can be found by
the method of
linear least-squares using a canned routine.
The unknown vector x contains the filter coefficients. The coefficients for
all of the
powers can be stacked into a single column vector. The simulated logs fill the
A matrix and the
19
CA 02483611 2005-12-08
target logs fill the b vector. Numerous simulated logs can be stacked so long
as there is a target
log for each. The system can then be written as shown in Figure 12. There are
p different
powers. Each of the s,,~2~-yn is a row vector containing a section of
simulated log. A total of m
different simulated logs. For s;,,, the log is just the conductivity log in
units of mhos/m. Each '
element of s;,3n is equal to the corresponding element of s,, ~ raised to the
power 3/2. The
higher powers are filled in a similar fashion. The length of each log section
is n, the same as
the length of each filter, so each f is a column vector of length n. There are
p different filters
corresponding to the powers 1 to (2p-1)/2. Each t, is a column vector that
contains a target
log, the desired output of the processing. The shifting by one sample is
necessary to
implement the convolution of the filter coe~cients with the simulated log.
The A matrix is partitioned into two parts. The bottom p rows in A are to make
the
filter sums equal to the homogeneous coefficients to satisfy ~; f,~ = c~. The
0 and I are row
vectors of length n filled with zeroes and ones; respectively. (It is possible
to build this
relationship into the top part of the system and reduce the number of unknowns
in each filter
by one, but this is notationally tedious, and the extra rows at the bottom
extra p rows
represent a small percentage of the total size of A.)
Each row of the A matrix together with the corresponding entry in the right-
hand side
vector can be weighted by an importance factor. The extra equations to ~; f,~
= cf can be
weighted heavily, since this is a requirement for the tool to read correctly
in a homogeneous
medium.
For p = 5, n = 401 (at quarter foot samples), 22 different Oklahoma profiles
at various
resistivity levels were used as the simulated logs. Each log was 150 feet (601
logging points).
The resulting A matrix is 13,227 by 2005 columns, over 100 MB at 32-bit single
precision.
The filters were designed by using the Oklahoma formation at numerous
conductivity levels.
. Figure 13 shows the deepest three-receiver measurement (but only the upper
one)
deconvolved with the new algorithm. The operating frequency was 8 kHz. The top
track
shows the error in the homogeneous boosting function versus true formation
conductivity.
Operatin~~ Frequency
The X-signal has been shown to be adversely affected by contrasts in magnetic
permeabilit~~. For this reason, some prior art deep signal processing uses the
X-signal only in
low resistivity formations, and does not use it in medium signal processing.
Reducing the
operating frequency could eliminate the remaining need for the X-signal in the
deconvolution.
CA 02483611 2005-12-08
The amount of skin effect goes as , f ~n but signal strength goes as , f .
Hence, reducing the
frequency reduces the sensitivity much more quickly than it reduces the skin
effect. If an
objective is to use the difference in the response at different frequencies to
measure some
unknown frequency dependence of the formation, the frequency range must be
small enough
S so as to achieve sufficient signal at the lowest frequency. Because there is
no particular
advantage to having a mufti-frequency tool, the preferred system includes a
dual-frequency
tool with both frequencies operating simultaneously. It is desirable to reduce
the operating
frequency, to reduce the need for the X-signal in low resistivities. On the
other hand, it is
desirable to raise the frequency to get more signals in high resistivities.
The simplest solution
is to pick two frequencies and measure both simultaneously. In a preferred
embodiment,
frequencies of approximately 8 and 32 kHz are used. This 4:1 frequency ratio
gives a 16:1
signal level ratio (for the same transmitter current). The 32 kHz gives 2.56
times as much
signal as a 20 kHz (for the same transmitter current. To simplify the signal
processing and to
further reduce the magnetic permeability effects, it may be desirable to
eliminate the X signal.
Operation
In operation, the first step is to convert the raw measurements into units of
apparent
conductivity. This is done using tool constants calculated during shop-
calibration. Next, the
sonde errors, also from shop-calibration, are subtracted from each
measurement. At this
point, there are 10 different apparent conductivity signals, both in-phase and
quadrature, at
the two operating frequencies.
The caliper information and mud resistivity measurements are convolved to
subtract
out the cave effect from each receiver measurement, using any suitable
algorithm.
There is one set of filter coefficients f j for each of the ten receiver
channels at both
frequencies, twenty sets in all. The deconvolution filters apply the depth
matching as well.
There will be two sets of twenty filters - one set for the one-foot vertical
resolution and
another set for the two-foot resolution. The present deconvolution
accomplishes the
following: skin-effect correction, shoulder-effect correction, depth
alignment, symmetrization
(in the absence of invasion), and resolution matching.
After deconvolution, the measurements from the lower receiver coils are
combined
with the corresponding measurements from the upper receiver coils. This
results in the six
different depths of investigation. The deepest four of the six will be fully
symmetric in depth
in the presence of invasion.
21
CA 02483611 2005-12-08
The six apparent conductivity curves depths of investigation are combined with
various weighting functions to produce the final 10, 20, 30, 60, 90 and 120
inch depths of
investigation. Optionally, the cross-sectional image of the invasion front can
be produced.
Auxiliary Measurements
In order to perform accurate environmental corrections, the foregoing
discussion
assumes that the following auxiliary measurements are present either in the
tool itself or
elsewhere in the tool string:
1. A z-axis accelerometer to perform speed correction to correct for. erratic
tool motion.
2. A mud resistivity sensor for accurate borehole correction.
3. A caliper to perform accurate borehole and cave correction of the shallow
curves of 10 and
inch depths of investigation.
4. A measurement of sonde temperature to compensate for any temperature-
dependent sonde
drifts.
5. A measurement of stand-off to improve the accuracy of the borehole
correction for the 10
15 and 20 inch depths of investigation.
While a preferred embodiment of the present invention has been shown and
described,
it will be understood that various modifications could be made to the
foregoing process and
apparatus without departing from the scope of the invention.
22
