Note: Descriptions are shown in the official language in which they were submitted.
CA 02037563 2000-04-14
An Electronic Measurement Apparatus Movable In
A Cased Borehole and Compensating for Casing
Resistance Differences ,
This invention was made with co-funded financial
support from: (a) U.S. Department of Energy (DOE) Grant
No. DE-FG19-88BC14243 entitled "Proof of Feasibility of
Thru Casing Resistivity Technology"; and (b) Gas Research
Institute (GRI) Contract No. 5088-212-1664 entitled "Proof
of Feasibility of the Through Casing Resistivity
Technology". The U.S. government and the GRI have certain
rights in this invention.
20
- 1 -
CA 02037563 2000-04-14
This invention provides improved methods and
apparatus for measurement of the electronic properties of
formations such as the resistivities, polarization
phenomena, and dielectric constants of geological
formations and cement layers adjacent to cased boreholes
and for measuring the skin effect of the casing present.
The terms "electronic properties of formations" and
"electrochemical properties of formations" are used
interchangeably herein. The methods disclosed in the
application provide for improved measurement accuracy in
the presence of corroded casing while performing
stationary measurements within cased wells. Furthermore,
apparatus and methods are disclosed which enable such
improved measurements to be performed while the apparatus
is being moved vertically in cased wells which is
typically desirable in the industry.
The oil industry has long sought to measure
resistivity through casing. Such resistivity
measurements, and measurement of other electrochemical
phenomena, are useful for at least the following purposes:
locating bypassed oil and gas; reservoir evaluation;
monitoring water floods; measuring quantitative
saturations; cement evaluation; permeability measurements;
and measurements through a drill string attached to a
drilling bit. Therefore, measurements of resistivity and
other electrochemical phenomena through metallic pipes,
and steel pipes in particular, are an important subject in
the oil industry. Many U.S. patents have issued in the
pertinent Subclass 368 of Class 324 of the United States
Patent and Trademark Office which address this subject.
The following presents a brief description of the
particularly relevant prior art presented in the order of
2
CA 02037563 2000-04-14
descending relative importance.
U.S. Patent No. 4,820,989 issued to the inventor, and'
U.S. Patent 4,882,542 issued to the inventor,
are henceforth to be referenced together as the
"Vail patents". These Vail patents predominantly describe
apparatus having two pairs of voltage measurement
electrodes which engage the interior of the casing, and
which have a calibration means to calibrate for thickness
variations and errors in the placements of the electrodes.
The detailed descriptions of several particular methods of
operation which are disclosed in the Vail patents have
certain limitations in.measurement accuracy due to "second
order errors" that will be described in detail in the
remaining portion of the application. Furthermore, while
the Vail patents do, briefly discuss taking data while
moving in the well, the application herein provides
detailed methods and apparatus for that purpose.
U.S. Patent No. 4,796,186 which issued on January 3,
1989 to Alexander A. Kaufman entitled "Conductivity
Determination in a Formation Having a Cased Well" also
mostly describes apparatus having two pairs of voltage
measurement electrodes which engage the interior of the
casing and which also have calibration means to calibrate
for thickness variations in the casing and for errors in
.the placements of the electrodes. In general, different
methods of operation are described in the Kaufman patent
compared to the Vail patents cited above. The particular
methods of 'operation in the Kaufman patent do not
thoroughly describe how~to eliminate all types of "second
order errors", nor does the Kaufman patent describe how to
build and operate an apparatus which takes data while
moving vertically in the well.
3
1~~''~'
U.S. Patent No. 4,837,518 which issued on June 6,
1989 to Michael F. Gard, John E. Kingman, and James D.
Klein, .assigned to the Atlant~.c Richfield Company,
entitled "Method and Apparatus for Measuring the
Electrical Resistivity of Geologic Formations Through,
Metal Drill Pipe or Casing", describes multiple voltage
measurement electrodes within a cased well which engage
the wall of the casing, henceforth referenced as "Arco's
patent". However, Arco's patent does not describe an
apparatus with two pairs of adjacent voltage measurement
electrodes and associated electronics which takes the
voltage differential between these two pairs to directly
measure electronic properties adjacent to formations.
Therefore, Arco's patent does not describe the methods and
apparatus disclosed herein.
USSR Patent No. 56,426, which issued on November 30,
1939 to L. M. Alpin, henceforth called the "Alpin
patent", which is entitled "Process of the Electrical
Measurement of Well casings", describes an apparatus
which has two pairs of voltage measurement electrodes
which positively engage the interior of the casing.
However, the Alpin patent does not have any suitable
calibration means to calibrate f or thickness variations
nor errors in the placements of the electrodes.
'Cheref ore, the Alpiri patent does not describe the methods
and apparatus disclosed herein.
U.S. Patent No. 2,729,784, issued on January 3, 1956
having the title of "Method and' Apparatus for Electric
Well Logging", and U.S. Patent No. 2,891,215 issued on
June 16, 1959 having the title of "Method and Apparatus
for Electric Well Logging", both of which issued in the
name of Robert E. Fearon, h.ezzcef orth called the "Fearon
patents", describe apparatus also having two pairs of
4
~~ '~ YJ ~i~ P.r
voltage measurement electrodes which engage the interior
of the casing. However, an attempt is made in the Fearon
patents to produce a "virtual electrode" on the Casing in
an attempt to measure leakage current into formation wYy ch
provides for methods and apparatus which are unrelated to
the Kaufman and Vail patents cited above. The Fearon
patents neither provide calibration means, nor do 'they
provide methods similar to those described an either the
Kaufman patent or the Vail patents, to calibrate for
thickness variations and errors in the placements of the
electrodes. Theref ore, the Fearan patents do not describe
the methods and apparatus disclosed herein.
Accordingly, an object of the invention is to provide
new and practical methods of measuring the resistivity of
geological formations adjacent to cased wells which
compensate far s~cand order errors of measurement.
Zt is yet another object of the invention to provide
new and practical methods of measuring the resistivity of
geological formations adjacent to cased wells which
compensate far second order errors of measurement and
which are also particularly adapted f or acquiring data
while moving vertically in the well.
And it is another object of the invention to provide
new, and practical apparatus capable of providing
measurements of the resistivity of geological formations
adjacent_to cased wells which compensate f or second order
errors of measurement and which also are capable of
acquiring data while moving vertically in the well.
And further, it is another object of the invention to
provide new and practical methods and apparatus capable of
measuring electrochemical phenomena through casing which
5
5
~~ ~,i~ E7 '~ '».J J cj
compensate f or second order errors of measurement.
And finally, it is another object of the invention to
provide new and practical methods and apparatus capable of
measuring electrochemical phenomena through casing which
compensate f ox second order errors of measurement and
which are also particularly adapted for acquiring data
while moving vertically. in the well.
Figure 1 is a sectional view of one preferred
embodiment of the invention of the Thru Casing Resistivity
Tool (TCRTJ.
Figure 2 shows QI vs. z~which diagrammatically
depicts the response of the to01 to different formations.
Figure 3 is a sectional view of a preferred
embodiment of the invention which shows how Vo is to be
measured.
Figure 4 is a sectional view of an embodiment of the
' invention which has voltage measurement electrodes which
are separated by different distances.
Figure 5 is a sectional view of an embodiment of the
invention which has electrodes which are separated by
different distances and which shows explicitly how to
measure Vo.
Figure 6 is a sectional view of an embodiment of the
invention which is adapted to take measurements while
moving vertically in the cased well.
The invention herein is described in two major
different portions of the specification. Tn the first
6
CA 02037563 2000-04-14
major portion of the specification the apparatus defined in
Figures 1, 3, 4, and 5 is described. Such
apparatus are perhaps ideally suited as stop-hold-and-lock
, apparatus for stationary measurements (although their
applications are certainly not limited to stop-hold-and-
lock situations). New and precise methods of measurement
are then disclosed in this first portion of the text which
provide for improved measurement accuracy of an apparatus
in the presence of corroded casing while performing
stationary measurements within cased wells. The second
major portion, of the specification herein is concerned
with providing both apparatus and methods which enable
such improved measurements to be performed while the
apparatus is being moved vertically in cased wells which
is typically desirable in the industry.
25
Fig. l shows a typical cased borehole found in an
oil field. The borehole 2 is surrounded with borehole
casing 4 which in turn is held in place by cement 6 in the
rock formation 8. An oil bearing strata 10 exists
adjacent to the cased borehole. The borehole casing may
or may not extend electrically to the surface of the earth
12. A voltage signal generator 14 (SG> provides an
A.C. voltage via cable 16 to power amplifier 18 (PA). The
signal generator represents a generic voltage source which
includes relatively simple devices such as an oscillator
7
to relatively complex electronics such as an arbitrary
waveform generator. ~.'he power amplifier 18 is used to
conduct A.C. current down insulated electrical wire 20 to
electrode A which is in electrical contact with the
casing. The current can return to the power amplifier
through cable 22 using two different paths. If' switch SWl
is connected to electrode B which is electrically grounded
to the surface of the earth, then current is conducted
primarily from the.power amplifier through cable 20 to
electrode A and then through the casing and cement layer
and subsequently through the rock formation back to
electrode B and ultimately through cable 22 back
to the power amplifier. In this case, most of the current
is passed through.the earth. Alternatively, if SW1 is
connected to insulated cable 24 which in turn is connect ed
to electrode F, which is in electrical contact with the
casing, then current is passed primarily from elect rode A
to electrode F along the casing for a subsequent return to
the power amplifier through cable 22. In this case, little
current passes~through the earth.
Electrodes C, D, and E are in electrical contact with
the interior of casing. In general, the current flowing
along the casing varies with position. For example,
current IC is flowing downward along the casing at
electrode C, current ID is,flowing downward at electrode
D, and current IE is flowing downward at electrode E. In
general, 'therefore, there is a voltage drop Vl between
electrodes C and D which is amplified differentially with
amplifier 26. And the voltage difference between
electrodes D and E, V2, is also amplified with amplifier
28. Wi.th~switches SWZ arid SW3 in their closed position as
shown, the outputs of amplifiers 26 and 28 respectively
are differentially subtracted with amplifier 30. The
voltage from amplifier 30 is sent to the surface via cable
8
J ,
f,l ~~ Cod
32 to a phase sensitive detector 34 (PSD?. The phase
sensitive detector obtains its reference signal from the
signal generator via cable 36. In addition, digital gain
controller 38 (GC? digitally controls the gain of
amplifier 28 using cable 40 to send commands downhole.
The gain cantroller 38 also has the capability to switch
the input leads to amplifier 28 on command, thereby
effectively. reversing the output polarity of the signal
emerging from amplifier 28 fox certain types of
measurements.
The total current conducted to electrode A is
measured by element 42. In the preferred embodiment shown
in Fig. 1, the A.C. current used is a symmetric sine wave
and therefore in the preferred embodiment, I i5 the 0-peak
value of the A.C. current conducted to electrode A. (The
0-peals value'of a sine wave is 1/2 the peak-to-peak value
of the sine wavo.l
In general, with SW1 connected to electrode B,
current is conducted through formation. For example,
current gI,is conducted into formation along the length
2L between electrodes C and E. However, if SWl is
connected to cable 24 and subsequently to electrode F,
then no current is conducted through formation to
electrode B. In this case, IC ID = IE sznce essentially
little current t1T is conducted into formation.
Tt should be noted that if SWl is connected to
electrode B then the current will tend to flow through the
formation and not along the borehole casing. Calculations
show that for 7 inch O.D. casing with a 1/2 inch wall
thic)tness that if the formation resistivity is 1 ohm-meter
and the formation is uniform, then approximately half of
the current will have flowed off the casing and into the
9
CA 02037563 2000-04-14
formation along a length of 320 meters of the casing. For
a uniform formation with a resistivity of 10 ohm-meters,
this length is 1040 meters instead.
One embodiment of the invention
provides a preferred method of operation
for the above apparatus as follows: "The first step in
measuring the resistivity of the formation is to "balance"
the tool. SW1 is switched to connect to cable 24 and
subsequently to electrode F. Then A.C. current is passed
from electrode A to electrode F thru the borehole casing.
Even though little current is conducted into formation,
the voltages V1 and V2 are in general different because
of thickness variations of the casing, inaccurate
placements of the electrodes, and numerous other factors.
However, the gain of amplifier 28 is adjusted using the
gain controller so that the differential voltage V3 is
nulled to zero. (Amplifier 28 may also have phase
balancing electronics if necessary to achieve null at any
given frequency of operation. ) Therefore, if the
electrodes are subsequently left in the same place after
balancing f or null, spurious effects such as thickness
variations in the casing do not affect the subsequent
measurements.
With SW1 then connected to electrode H, the signal
generator drives the power amplifier which conducts.
current to electrode A which is in electrical contact with
the interior of the borehole casing. A.C. currents from 1
amps o-peak to 30 amps o-peak at a frequency of typically
1 Hz is introduced on the casing here. The low frequency
operation is limited by electrochemical effects such as
polarizatrion phenomena and the invention can probably be
operated down to .1 Hz and the resistivity still properly
measured. The high frequency operation is limited by skin
CA 02037563 2000-04-14
depth effects of the casing, and an upper frequency limit
of the invention is probably 20 Hz for resistivity
measurements. Current is subsequently conducted along the
casing, both up and down the casing from electrode A, and
some current passes through the brine saturated cement
surrounding the casing and ultimately through~the various
resistive zones surrounding the casing. The current is
then subsequently returned to the earth's surface through
electrode H."
Therefore, in a preferred method of operation,
first the tool is
"balanced" for a null output from amplifier 30 when SW1 is
connected to cable 24, and then the departure of the
signal from null when SW1 is instead connected to
electrode B provides a measure of the leakage current
into formation. Such a method of operation does not
automatically eliminate all "second order errors of
measurement". An improved method of operation is
described in a later section of this application which
does automatically eliminate these "second order errors of
measurement".
Fig. 2
shows the differential current conducted into formation
6I for different vertical positions z within a steel cased
borehole. z is defined as the position of electrode D in
Fig. 1. It should be noted that with a voltage applied to
electrode A and with SW1 connected to electrode B that
this situation consequently results in a radially
symmetric electric field being applied to the formation
which is approximately perpendicular to the casing. The
electrical field produces outward flowing currents such as
(iI in Fig. 1 which are inversely proportional to the
re~istivity of the formation. Therefore, one may expect
11
f ~ '-'.! ~ G
discontinuous changes in the current t1I at the interface
between various resistive zones particularly at oillwater
and oil/gas boundaries. Far example, curve (a) in Fig. 2
shows the results from a uniform formation with
resistivity p1 . Curve tb) shows departures from curve
(a) when a formation of resistivity p2 and thickness T2 is
intersected where p2 is less than pl. And curve tc)
shows the opposite situation where a formation is
intersected with resistivity 'p3 which is greater than
pl which has a thickness of T3. It is obvious that under
these circumstances, ~I3 is less than ~I1, which is
less than ~I2.
Fig. 3 shows a detailed method to measure the
parameter Vo. Electrodes A, B, C, D, E, and F have been
defined in Fig. 1. All of the numbered elements 2 through
40 have already been defined in Fig. 1. In Fig. 3, the
thickness of 'the Casing is T1, the thickness of the cement
is TZ, and d is the diameter of 'the casing. Switches SW1,
SW2, and SW3 have also been defined in Fig., 1. In
addition, electrode G is introduced in Fig. 3 which is the
voltage measuring rAference electrode which is in
electrical contact with the surface of the earth. '.this
electrode is used as a reference electrode and conducts
little current to avoid measurement errors associated with
current f low.
In addition, SW4 is introduced. in Fig. 3 which allows
the connection of cable 24 to one of the three positions:
to an open circuit; to electrode G; or to the top of the
borehole casing. And in addition in Fig. 3, switches SW5,
SW6, and SW7 have been added which can be operated in the
positions shown. (The apparatus in Fig. 3 can be operated
in an identical manner as that shown in Fig. 1 provided
that switches SW2, SW5, SW6, and SW? are switched into the
12
CA 02037563 2000-04-14
opposite states as shown in Fig. 3 and provided that SW4
is placed in the open circuit position.)
With switches SW2, SWS, SW6, and SW7 operated as
shown in Fig. 3, then the quantity Vo may be measured.
For a given current I conducted to electrode A, then the
casing at that point is elevated in potential with respect
to the zero potential at a hypothetical point which is an
"infinite" distance from the casing. Over the interval of
the casing between electrodes C, D, and E in Fig. 3, there
exists an average potential over that interval with
respect to an infinitely distant reference point.
However, the potential measured between only electrode E
and electrode G approximates Vo provided the separation of
electrodes A, C, D, and E are less than some critical
distance such as lO.meters and provided that electrode G
is at a distance exceeding another critical distance from
the casing such as 10 meters from,the borehole casing.
The output of amplifier 28 is determined by the voltage
difference between electrode E and the other input to
the amplifier which is provided by cable 24. With SWl
connected to electrode B, and SW4 connected to electrode
G, cable 24 is essentially at the same potential as
electrode G and Vo is measured appropriately with the
phase sensitive detector 34. In many cases, SW4 may
instead be connected to the top of the casing which will
work provided electrode A is beyond a critical
depth.
For
the purposes of precise written descriptions of the
invention, electrode A is the upper current conducting
electrode which is in electrical contact with the interior
of the borehole casing; electrode B is the current
conducting electrode which is in electrical contact with
13
CA 02037563 2000-04-14
the surface of the earth; electrodes C, D, .and E are
voltage measuring electrodes which are in electrical
contact with the interior of the borehole casing;
electrode F is the lower current conducting electrode
which is in electrical contact with the interior of the
borehole casing; and electrode G is the voltage measuring
reference electrode which is in electrical contact with
the surface of the earth.
Furthermore, Vo is called the local casing potential.
~An example of an electronics difference means is the
combination of amplifiers 26, 28, and 30. The
differential current conducted into the formation to be
measured is DI. The differential voltage is that
voltage in Figure 1 which is the output of amplifier 30
with SW1 connected to electrode B and with all the other
switches in the positions shown.
Fig. 4
is nearly identical to Fig. 1 except the electrodes C and
D are separated by length Ll, electrodes D and E are
separated by L2, electrodes A~and C are separated by L3
and electrodes E and F are separated by the distance L4.
In addition, rl is the radial distance of separation of
electrode B from the casing. And z is the depth from the
surface of the earth to electrode D. Fig. 5 is nearly
identical to Fig. 3 except here too the distances L1~ L2'
L3, L4, rl, and z are explicitly shown. In addition, r2
is also defined which is the radial distance from the
casing to electrode G. As will be shown explicitly in
later analysis, the invention will work well if Ll and L2
are not equal. And f or many types of measurements, the
distances L3 and L4 are not very important provided that
they are not much larger in magnitude than Ll and
L2.
14
An improved method of operation of a preferred
eznbodimewt of 'the invention is now described in terms of
Figure 1. Using this "Preferred Method of Operation"
disclosed herein, the apparatus can be placed in any one
of three states: tl) the Preferred Measurement State;
(2) the Preferred Null State ; and (3) the Preferred
Calibration State.
The three states defined in this Preferred Method of
Operation of the invention are therefore defined as
f of lows :
(1) The Preferred Measurement State of the apparatus
in Figure 1 is defined by the following configuration:
SW1 is connected to electrode B; switch SW2 is closed; arid
switch 5W3 is closed.
(2) The Preferred Null State of the apparatus in
Figure 1 is defined by the following conf9.guration: SWl
is connected to cable wire 24 and therefore is
electrically connected to electrode F; switch SW2 is
closed; and switch SW3 is closed.
(3) The Pref erred Calibration State of the apparatus
in Figure 1 is defined by the following configuration:
SW1 is connected to cable 24 and therefore is electrically
connected to electrode F; switch SW2 is open; and switch
SW3 is closed.
. The purpose of the, following analysis is to determine
a method of operation to acquire the experimental quantity
6T in Figure 1 which is relatively accurate, which is
relatively unaffected by even substantial thickness
variations in the casing, and which is~ not materially
affected by inaccurate placements of the electrodes.
r3 ~.~ ~..1 e~
Therefore, assume that the gains of amplifiers 26, 28, and
'30 in Figure 1 are respectively.given by al, a2, and a3.
Furthermore, assume 'that the resistance of the casing in
ohms between electrode C and D is given by R1; and that
the resistance of the casing in ohms between electrodes D
and E is given by R2. In 'the following analysis, for the
purposes of simplicity only, it is assumed that: (a) there
is no phase shift between amplifiers; (b) there is no
phase shift between the voltages appearing on the casing
and the~current conducted through the earth (caused by
polarization effects, skin effects, or other
electrochemical processes); and (c) that therefore 'the
magnitudes representative of the amplitudes of low
frequency A.C. quantities may be used in the following
simplified analysis.
The average resistance of the casing between
electrodes D and E is defined as the quantity RA, which is
given as the follows:
RA - y R1 + R2 1 / 2 Equation 1.
Therefore, there is a departure from average
resistance of the first section of the casing between
electrodes C and D defined as ~ Rl such that:
Rl - RA ~. pRl Equation 2.
Furthermore, there is a departure from the average
resistance of the second section of the casing between
electrodes D and E defined as Q R2 such that:
R2 _ RA .~ nR2 Equation 3.
16
u, ~~f '~ ;~ t'~ y ~y
'~~ e.i~ 3 .~ i5 e,~
In the Preferred Null State of the apparatus, a
current called the Null Current (defined as "IN") is
passed along the casing between electrodes C and E. Since
relatively little current is expected t o flow 'through
formation in this slat e, then essentially the same current
IN flows between electrodes C and D and between electrodes
D and E. Therefore, the output voltage from amplifier 30
in this situation defined as VN, which is given by the
following:
to
VN _ a3 ~ a2 IN ( RA + QR2 y -- al IN ( RA + aRl ) 1
Equation 4.
Re-arranging terms in Equation 4, one obtains:
VN _ a3,'y IN Equation 5.
The quantity Y in Equation 5 is given by the
following algebraic formula:
Y = ~ a2 ( RA + QR2 ) - al ( RA + ~R1 ) ?
Equation 6.
In the Preferred Measurement State of the apparatus,
a "Total Measurement Current", defined as IT, is passed
between electrode A .and electrode B on the surface of the
earth. Only a portion of that Total Measurement Current
is conducted downward along the casing between electrodes
C and D, and~that portion is called simply the
"Measurement Current", defined as the quantity IM. If TM
is passing between electrodes C and D, then a certain
current will leak off the casing between electrodes D and
E which is defined as the quantity d:i2. Therefore, the
17
current passing downward at electrode E, which is the
quantity IE, is given by the following:
TE = T~ -- 8 i2 Equation 7 .
Therefore, the output voltage from amplifier 30 in this
situation is defined as.VM. which in a lumped component
model approximation is given by the following:
VM = a3 C a2 ( IM - &i2) ( R~ -~- 4R2) -- al IM ( R~ -~ dRl)1
Equation B.
Equation 8 simplifies to the following:
VM _ a3 Y I~ _ a2 a3 g i2 ( RA -h L1R2 )
Equation 9.
In the Preferred Calibration State of the apparatus,
the.calibration current, ICS, is passed along the casing
between electrodes C and E. Since relatively little
current is expected to flow through f ormatian in this
state, then essentially the same current TCA flows between
electrodes C and D and between electrodes D and E. In the
Preferred Calibration State, SW2 is open, so that the
output from amplifier 30 in this state, VCS, is given by
the following:
VCA - " al a3 ICA ( R,~ -~ 4R1 ) / D
Equation 10.
Here, the parameter D is a gain reduction
parameter. or divider factor, to avoid saturation of
amplifier 30 if SW2 is open,
la
~~'~':~~~r
although the means to do so is not explicitly shown in
Figure 1. Basically, when SW2 is open, then the output of
amplifier 30 is divided by the parameter D which was found
to be useful during field tests. In this embodiment, the
gain controller 3n in Figure 1 has at least the ability to
open and close switches SW2, SW3, and t o divide the output
of amplifier 30 by the factor D.
There are three fundamentally different independent
measurements performed which are summarized in the
f ollawing equations: Equation 5, Equation 9, and
Equation 10. The gains al, a2, and a3 are assumed to be
known to an accuracy of 0.1%. The following currents are
measured during the various different states: IN, IM, and
IUD. Therefore, Equations No. 5,,9, and 10 contain
essentially three fundamental unknowns: di2, Rl, and R2.
Three equations with three unknowns always have a unique
solution. One such solution used in practice that
minimizes certain types of required measurement accuracies
is described in the following.
A convenient parameter to calculate for the analysis
herein is called AV, which is defined as follows:
A V - V~ - VN Equation 11.
Using VM from Equation 9, and VN .from Equation 5,
then the quantity AV is given as follows:
A V - a3 Y ( IM - TN ) - a2 a3 fi i2 t RA + a.R2 )
Equation 12.
19
f., r,
~.~ ~ ~i '3 ~J P~a
Using Equation 10, and solving for the quantity RA:
RA _ ~ - D VCA / ( al a3 ICA ) ~ - 6 R.~
Equation 13.
Substituting RA from Equation 13 into Equation 12,
then the quantity ~V becomes:
aV - ( a2 / al ) ( d i2 ! FICA ) ~ VCA CTerm A~1
+ a3 y ( IM - IN) GTerm B3
- a2 a3 d i2 ( D R2 - ~ R1 > CTerm C:7
Equation 1~.
Equat:iAn l~ is the central result of this analysis.
'.Germs A, B. and C appear in sequence in Equation 14 and
are identa..fied by the square brackets adjacent to each
respective term. Term A describes useful information
related to measuring the leakage current flowing into
formation. Term B is a first order error term which is
present if the measurement current IM is not equi alent to
the null current IN and if there are different resistances
in different portions of the casing. Term C is the second
order error term which is due to the product of the
difference in resistances between different sections of
casing times the leakage current f lowing into formation.
Term C is called a second order term because it is the
product of two quantities which can each be very small
under ideal circumstanices.
20
~,J f,. ~fi ,~
~~ ~ ~ ...1 fj r3
For the purpose of showing the relative numerical
sizes of Terms A, B, and C, and their relative importance,
these terms are now calculated for 'the following values:
al. _ a2 -. A
a3 - 10
D - 10
Equation 15
pRl - - O.1 RA
p R2 - -!- 0 . 3 RA
- ~cA - . ~o
Furthermore, for the purpose of conveying the
importance of the first order error Term B above, it is
assumed that the currents T~ and 1N are almost, but not
quite equal, as follows:
1M - IN - - 0.01 To Equation 16.
Equation 10 is first used to calculate the quantity
VCA ~.n Term A o:~ Equation 14; then Equations 15 and 16 are
substituted into Equation 14; and using Equation 6 f or the
quantity y ; then Equation 14 becomes:
Term A - --0:90 A a3 8iZ RA Equation 17.
Term B - -0.004 A a3 Io RA Equation ld.
Term C - -0.40 A a3 di2 RA Equation 19.
35.
21
~~~~r
Tn this e~cample, the second order error term, Term C
is significant. Keeping in mind that Term A has the
information desired related to current leakage from the
casing:
Term C l Term A - 0.44 Equation 20.
The second order error term causes a 44% error in
computing the leakage current if onl r.~erm A of Equation
14 is used to calculate leakage current into formation.
Theref ore, it is concluded that methods for compensation
of second order errors are important for improved
measurement accuracy.
~lith a leakage current of ,about 30 milliamps, and
with current T~ of 5, amps, then the ratio of Terms ~i to A
becomes:
Term S / Term A - 0.74 Equation 21.
Therefore, if ythe currents TM and TN are equal to
only 1% as in Equation 16, 74% measurement errors are to
be expected in this example if onl Term A in Equation 14
is used to calculate the leakage current flowing into
formation.
The result in Equation 21 also points up an important
aspect of this analysis. Tf the currents TM and TN were
different by 50% as is typically the case if the total
currents delivered to the casi_~ _throuah current meter 42
were kept_constant for different switch positions of SW1,
then errors on the order of at least 25 times the size of
the quantity to be measured would be present in the
example cited. To compensate f or such large errors, the
resistances R1 and R2 would have to be measured to an
22
accuracy approaching 5 significant decimal places far the
required measured accuracy of the leakage currentl This
place accuracy is difficult to achieve because of
cross-talk in the wirelines, magnetic pick-up, and other
5 effects that are to be discussed in another related
application. Hy contrast, the method of operation
described above only requires consistent measurement
accuracy of the various quantities to typically 3 decimal
places which is much easier to achieve in practice.
Therefore, in practice in one embodiment of the .
invention, the currents IM and I~ are to be made equal to
an accuracy of one part in one thousand (0.1%) to give
acceptable measurement errors of the leakage current to
accuracies better than 10% if Term B in Equation 14 is to
be neglected. during interpretation of the data.
Consequently, in practice to avoid first order errors of
measurement if Term B is to be neglected
tand.al = al = A):.
Absolute Magnitude of (IM - IN) '~ .001 Ta
Equation 22.
Equation 14 is the central result and is used to
interpret the data. Now that the importance of the
various terms have been established, the following logic
is used to put Equation 14 in a farm that is used in
practice in one embodiment of the invention. In
particular, if the gains of amplifiers 26 and 28 are
equal, then the following is used to simplify Equation 14:
al _ a2 _ ~, Equation 23.
23
;.~'~~e~~a.J~:9e.~
Substituting Equation 23 into Equation 6 for the
quantity Y gives the following that may be used to
simplify Term B in Equation 14:
y = A ( ~R2 - ~R1) Equation 24.
Term C in Equation 14 can be put into another form
more useful for this analysis if Equation 23 is
substituted into Equations 5 and 6 thereby providing:
( ~R2 - 4R1 ) - VN / ( a3 A IN )
Equation 25.
Therefore, Equation 24 is used to simplify Term B in
Equation 14, and Equation 25 is used to simplify Term C in
Equation 14r 'thereby providing:
0V - di2 C ( D VCA f xCA ) - ( VN / TN ) 1
+ a3 - A C ( T~ - IN ) ( ~R2 - ~Rl ) 1
Equation 26
The first line of Equation 26 has the corresponding
Terms A and C from Equation 14. The second line of
Equation 26 corresponds to Term B of Equation 14. Please
notice that this second term is the mathematical product
of the difference in resistances between various sections
of casing times the difference between the measurement and
null currents, which in general is a first order error
term that is very large unless the measurement and null
currents are deliberately chosen to be equal for reasons
discussed above.
24
~0~°~
Tn a. preferred method of operation the invention,
the experimental parameters may be set such thats
IM - IN - ~TCA " To Equation 27.
In practice, these currents are made equal by
demanding that the voyage from amplifier 3U with SW2
open, and SW3 closed, be the same voltage f or both
possible states of SWI. For each position of SWl,
suitable adjustments are made on the,output voltage of the
signal generator 14 in Figure 1 such that the current
flowing between electrodes C and D is always the.same for
each position of SW1 in this preferred embodiment of the
invention.
20
If Equation 27 is exactly valid, 'then Term B in
Equation 26 is zero. Using Equata.on 11 to eliminate the
quantity ~V :Ln.Equation 26, then Equation 26 becomes the
following:
8 i 2 - T o ( VM - VN ) / ( D VCA -- VN )
Equation 28.
This is the primary result which,provides the
quantity di2 which is not affected by second order errors
of measurement caused by different resistances in adjacent
regions of the casing. The quantity di2 is also not
affected by first order errors of measurement.
Therefore, at least three measurements of different
quantities are necessary to eliminate significant
measurement errors. and to provide accurate results. In
the case shown above, three independent~measurements
result 3n measuring three experimentally independent
quantities: di2, Rl, and R2. Put another way, first and
_sP_cond compensation measurements are performed to correct
a third measurement of the current leakin into formation
from 'the casino to obtain accurate results.
Furthermore. the apparatus has bean used in three
different configurations providing a first compensation
means (Preferred Null State) and then a second
compensation means (Preferred Calibration State) to obtain
accurate measurements of the leakage current (in the
Preferred Measurement State).
It is to be briefly noted that Equation 6 in U.S.
patent No. 4,796,186 is directed toward eli.minatinc~
various types of measurement errors. That equation has
been analyzed using the above concepts. In the language
contained herein, Equation 6 in U.S. Patent No. 4,796,186
does in fact eliminate first order errors of measurement
like Term :B in Equation 14. However. it does not
eliminate second order errors of measurement like
Term C in Equation 14. This does make sense since U.S.
patent No. 4,796,186 emphasizes making two independent
measurements to obtain data (one compensation step, and
one measurement step). The application herein has shown
that ~sacond order measurement errors are of importance
which therefore requires a method having two independent
compensation steps and one final measurement step to
eliminate those second order measurement errors.
U.S. Patent No. 4,820,989 defines a method of
analysis which gives the resistivity of adjacent
formations. See particularly Equations 6, 8, and Equation
9 (as corrected in a Certificate of Correction). However,
in'that U.S. patent, the quantity ~I is discussed which
26
is the combined current loss between both electrodes C - D
and between electrodes D ° E. However, here, this method
of operation distinguishes between the current loss
between electrode C and D which is bil, and the current
loss between electrodes D and E which is di2. Therefore,
using a lumped component model approx:i.mation:
d:i2 - Q.1 - d~il Equation 29.
To obtain the resistivity of the adjacent formation,
the following analysis is needed. The contact resistance
Rc of a cylindrical electrode of length t and of diameter
d in contact with a formation of resistivity p is given
theoretically' by the following (Earth Resistances, G. I°.
Tagg. Pitman PubJ.is)i~:ng Corporation, N.Y. 194, pg. 96):
Rc = p ~ Ln (~t/d) J /(2 ~r t) Equation 30.
.- p C
The geometric constant C is defined from theoretical
considerations in Equation 30. The experimentally
measured value~of the contact resistance, Rc, is given by
the following:
Rc - Vo / S~i2 Equation 31.
Vo in Equation 31 is given by the measured potential
voltage on the casing for the current Io defined in
Equation 28. Therefore, the resistivity p is
given by the following:
p = Vo / ( C ai2 ) Equation 32.
27
It is most convenient to define a new parameter K
where:
K - 1 f C Equation 33.
Then Equation 32 becomes:
p - K Vo / ai2 Equation 34.
Using the definition of Rc from Equation 31, then
Equation 34 finally becomes:
p - K RC Equation 35.
Equation 35 provides the final result used to
interpret the data for the preferred method of operation
herein. Please notice that the parameter K has units of
meters which is sensible because RC has units of ohms, and
resistivity is measured in the units of ohm-meters. It
should be noted that any calculated values of the
parameter K from Equations 30 and 33 for a given diameter
of casing and separation of voltage measurement electrodes
would rely upon the validity of the argument that the
section of caging of length. t may be approximated by an
ellipsoid of revolution described by Tagg. It is
therefore to be expected that the any value of K
calculated from such an analysis may not be highly
accurate because of this approximation. It is reasonable,
however, to expect that the parameter K could be
empirically determined with considerable accuracy.
Calibrating the apparatus against a formation with known
resistivity adjacent to a casing that was measured before
28
~~~'~K~~.
the casing was sot is sufficient for the purpose of
empirically determining the parameter K. From Equations
30 and 33, it is expected that the parameter K is to be a
function o~ the diameter of the pipe. From those two same
equations, the parameter K is also expected to be a
function of the spacing between the voltage measurement
electrodes.
The second major portion of the specification is used
to describe Figure 6 which presents an apparatus adapted
to take data while moving which allows utilization of the
preceding fundamental analysis. A preferred apparatus
optimally adapted to take data while moving which uses the
above fundamental methods of analysis requires a machine
which simultaneously -acquires data analocrous to that
provided above in the Preferred Measurement State; the
Preferred Null State; and in the Preferred Calibration
State. The apparatus in Figure 6 is adapted to acquire
data in the analogous Preferred l~leasurement State at a
first frequency of operation, called ,"F(1)" (perhaps 1 Hz
for example); is adapted to simultaneously acqux.re data a.n
the analogous Preferred Null State of the apparatus at a
second frequency of operation, called "F(2)" (perhaps 2 Hz
for example); and that is adapted to simultaneously
provide data in the analogous Preferred Calibration State
which ultimately is used to )seep the current flowing
betweem electrodes C and D at the two frequencies of
operation to be the same. as required by the conditions of
Equation 27 above.
For the purpose of logical introduction, the elements
in Figure 6 are first briefly compared to those in Figures
1-5. This introduction also serves to identify the
various legends used in Figure 6. hlements No. 2, 9, 6,
8, and 10 have already been defined. Electrodes A, B, C,
29
r~e3 a~c)~.~eJ
D, E, F, G and the distances L1, L2, L3, and Lq have
already been described. The quantities ail and ai2 have
already been defined in the above text. Amplifiers
labeled with legends A1, A2, and A3 are analogous
respectively to amplifiers 26, 28, and 30 defined in
Figures 1, 3, ~1, arzd 5. Tn addition, the apparatus i11
Fig. 6 provides for the following:
(a) two signal, generators 7.abeled with legends "SG 1
at Freq F(1)" and "SG 2 at Freq F(2)'~';
(b) two power amplifiers labeled faith legends "PA 1"
and "PA 2";
(c) a total of 5 phase sensitive detectors labeled
with legends "PSD 1", "PSD 2", "'PSD 3", "PSD 4", and
"PSD 5", which respectively have inputs for measurement
labeled with legends "STG", which have inputs for
reference signals labeled with legends "REF", which have
oLitpLlts defined by lines having arrows pointing away from
the re~spoct~.ve units, and which are capable oL reiectinc~
_a11 signal -voltages at freauencies which are not ecLzal to
' that t~rovided by the respective reference signals;
(d) an "Error Difference Amp" so labeled with this
legond in Fig. 6;
(e) an instrument which controls gain with voltage,
typically called a "voltage controlled gain", lJh:iCh i.S
. labeled with legend "VCG";
(f) an additional current conducting electrode
labeled with legend "11" (which is a distance L5 - not
shown - above electrode A).
(g) an additional voltage measuring electrode
labeled with legend "J" (which is a distance L6 - riot
shown - below electrode F).
(h) current measurement devices, or meters, labeled
with legends "T1" and "T2";
(i) and differential voltage amplifier labeled with
legend "A9" in Fig. 6.
r
The apparatus works as follows. Analogously to the
'°Preferred Measurement State", SG 1 provides the basic
signal voltage to the VCG which determines the output
level of PA 1 which therefore conducts current at
frequency F(1) through meter I1, through cable 4~ to
electrode A, and then betcaeen electrodes A and B through
the geological formation, and from electrode t3 through
cable ~G back to PA 1. Therefore, a voltage drop at the
frequency F,.(.1) appears along the casing between electrodes
C-D and D-E.~::.',~tmplifier A1 takes' the difference in voltage
between electrodes C-D; amplifier A2 takes the difference
in voltage between ,electrodes D-E; and amplifier A3 takes
the voltage difference, between the outputs in voltage from
respectively amplifiers A2 and A1 which is ser7t uphole to
PSD 1 and PSD 3 by cable 98. The voltage rneasured :by
PSD 1 is that analogous to VM above.
Analogously to the "Preferred Null State", SG 2
provides the basic signal voltage which determines the
output level of. PA 2 which therefore conducts current at
frequency F(2) tYrrough meter I2, through cable 50 to
electrode H, through a section of casing to electrode F,
and th en back through cable 52 to PA 2. Therefore, a
voltage drop at the frequency F(2) also appears along the
casing between electrodes C-D and D-E. Amplifier A1 takes
the difference in voltage between electrodes C-D;
amplifier A2 takes the difference in voltage between
electrodes D-E; and amplifier A3 takes the voltage
difference between the outputs in voltage from
respectively amplifiers A2 and A1 which is sent uphole to
PSD 1 and PSD 3 by cable 98. The voltage measured by
PSD 3 is analogous to VN defined above.
Analogously to the Preferred Calibration State, the
voltage appearing across electrodes C-D is amplified by
31
~~'~~~~
amplifier A1 which 15 then Sellt via cable 5~ to both PSD 2
and PSD ~. PSD ~1 measures a quantity related to the
current at frequency F(1) flowing through the casing
between electrodes C-D. PSD 2 measures a quantity related
to the current at frequency P'(2) flowing through the
casing between electrodes C-D. The outpLZts from PSD 2 and
PSD ~ are sent by cables respectively 56 and 58 to the
Error Difference Amp, which provides an error signal to
the VCG through cable 60. The voltage provided to the VCG
by SG 1 on cable 62 provides tl~ze:basic level of current
passing through meter I1. However, the error voltage
provided to the VCG by cable 60 provides adjustments in
the current passing through I1 such that the currents at
freawencies F(1) and at F(2) passing through the section
of casing between electrodes C and D are eaual to ttie
level of ~~recision defined in Equation 22 above.
To get resistivity, the potential voltage Vo is
needed. 'fherefore,~electrode J provides one of two input
voltages to amplifier A~, the other coming from electrode
G. The differential voltage output of amplifier A4 is
sent to PSD 5 via cable 69. PSD 5 obtains its reference
voltage from SG 1 at frequency F(1). Vo is measured at
the frequency F(1) by PSD 5. Therefore, sufficient
information is provided to the recording and computing
system to calculate the resistivity in a form analogous to
Equation 32 above.
For the purposes of clarity please notice that
SG 1 provides an output voltage at frequency F(1) which is
connected to voltage node 66 which in turn is connected to
the reference inputs of PSD 1, PSD4, and PSD 5.
Furthermore, please also notice that SG 2 provides a
voltage output at frequency F(2) which is connected to
voltage node 68 which in turn is connected to the
32
reference inputs of PSD 2 and PSD 3.
Digital Recording System 70 is labeled with legend
"DTG REC SXSTEM°'. It is capable of acquiring data, is
capable of performing computations with an internal
computer, and is capable of providing final processed data
outputs in the form of "logs" as is characteristic of the
well lagging industry. The output of PSD 5 is connected
to the input of the Digital Recording System with wire 72.
Similarly the output of PSD 1 is.connected to the input of
the Digital Reco>~ding System with mire 74. However, to
avoid unnecessary complexity in Figure 6, only the segment
of the wire emerging from PSD 1 is shown with an outgoing
arrow (labeled with 74) and only that portion of the wire
immediately adjacent to the input of Digital Recording
System is shown with an input arrow (also labeled with
74). xn general, an outgoing numbered arrow on a phase
sensitive detector is to represent a wire attach ed to its
output. An inward pointing arrow into the Digital
Recording System is to represent a wire attached to its
izyput. The outgoing arrow from a phase sensitive detector
having a given number is to be connected with a~wire to
the corresponding input arrow on the Digital Recording
System. Therefore, the output of PSD 3 is connected to
the input of the Digital Recording System with wire 76.
The output of PSD 2 is to be connected to the input of the
.Digital Recording System with wire 78. The output of PSD
4 is to.be connected to the input of the Digital Recording
System with wire 80. Tn addition, wire 82 connects the
output of amplifier A2 to an input of amplifier A3. Many
recent phase sensitive detectors have digital outputs.
Such digital outputs are ideal to provide digital data to
the computer within the Digital Recording System. On the
other hand, many older phase sensitive detectors have oz~zly
analogue outputs. In such a case, the corresponding
33
,.~ , 1" ~,, ., ,
~~.~~y~r~
inputs of the Digital Recording System must obviously be
analogue-to-digital converters to communicate with the
internal computer withiau the Digital Retarding System.
That internal computer is capable of performing standard
mathematical computations. The data so provided yield
sufficient information to calculate a leak age current
analogous to that c'lefined in Equation 2B. The exact
algebraic equation appropriate for the apparatus depends
upon the gains chosen for amplifiers A1, A2, A3, and A4,
but an equation closely analogous to Equation 2B for the
moving system can easily be determined. The computer is
then able to calculate the leakage current flowing into
formation. The infarma-tion provided to the Digital
Recording System is also sufficient to determine the
potential voltage Vo. The computor is then able to
calculate th a resistivity of the adjacent geological
formation using Equation 3~. The resistivity vs. ctepth in
the well is then provided in the form of a typical "lag"
characteristic of the well lagging industry.
To briefly to summarize: PSD 1 measures a quantity
analogous to VM; PSD 3 measures a quantity analogous to
VN; PSD 2 and PSD 4 are used to keep the current constant
at the two frequencies between electrodes C and D; and
PSD 5 measures Vo. The recording and computing system
then computes the current leakage into formation, the
local casing potential voltage, and the resistivity of the
adjacent geological formation while the system is being
drawn vertically upward in the well.
Uther components typically used in the wireline
industry are to be provided with the apparatus which are
not shown, but which include: depth control
instrumentation; logging vehicles and w:irelin es; pressure
housings for the downhole instrumentation, etc. It is
39
~~':~n:~~
assumed that the accustomed art in the industry :is to be
implemented in th a invention for the blue-print type
design of the invention and for the fabrication of the
invention. Typical methods of ''de-bugging" the apparatus
are to be employed including empirically determining the
maximum vertical draw rate of the apparatus by
successively moving at faster and faster~speeds until a
critical speed is reached whereby tl~e results no longer
agree with results at stationary or slower speeds of
movement.
While the above description contains many
specificities, these should not be construed as
limitations on the scope of the invention, but rather as
exemplification of preferred embodiments thereto. As has
been briefly described, there axe many possible
variations. Furthermore, the apparatus and methods
described may be used for at least the following purposes:
locating bypassed oil and gas; reservoir evaluation;
monitoring water floods; measuring quantitative
saturations; cement evaluation; permeability measurements;
and measurements through a drill string attached to a
drilling bit. Accordingly, the scope of tlae invention
should be determined not only by the embodiments
illustrated, but by the appended claims and their legal
equivalents.
35
