U.S. patent application number 15/031808 was filed with the patent office on 2016-08-25 for magnetic gradient and curvature based ranging method.
This patent application is currently assigned to Schlumberger Technology Corporation. The applicant listed for this patent is SCHLUMBERGER TECHNOLOGY CORPORATION. Invention is credited to Andrew G. Brooks, Leon Ceh, Graham A. McElhinney, Kenneth Stenerson.
Application Number | 20160245072 15/031808 |
Document ID | / |
Family ID | 52993569 |
Filed Date | 2016-08-25 |
United States Patent
Application |
20160245072 |
Kind Code |
A1 |
Brooks; Andrew G. ; et
al. |
August 25, 2016 |
Magnetic Gradient and Curvature Based Ranging Method
Abstract
Methods for determining a distance from a drilling well to a
magnetized target well include acquiring magnetic field
measurements from the drilling well. The acquired magnetic field
measurements are made at a plurality of spaced apart locations in
the drilling well. The acquired magnetic field measurements are
processed to obtain a ratio including at least one of the
following: (i) a ratio of a magnetic field intensity to a first
spatial derivative of a magnetic field, (ii) a ratio of a magnetic
field intensity to a second spatial derivative of a magnetic field,
and (iii) a ratio of a first spatial derivative of a magnetic field
to a second spatial derivative of the magnetic field. The ratio (or
ratios) is then processed to obtain the distance from the drilling
well to the magnetized target well.
Inventors: |
Brooks; Andrew G.; (Tomball,
TX) ; McElhinney; Graham A.; (Inverurie, GB) ;
Ceh; Leon; (Calgary, CA) ; Stenerson; Kenneth;
(St. Albert, CA) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
SCHLUMBERGER TECHNOLOGY CORPORATION |
Sugar Land |
TX |
US |
|
|
Assignee: |
Schlumberger Technology
Corporation
Sugar Land
TX
|
Family ID: |
52993569 |
Appl. No.: |
15/031808 |
Filed: |
October 23, 2014 |
PCT Filed: |
October 23, 2014 |
PCT NO: |
PCT/US2014/062006 |
371 Date: |
April 25, 2016 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
61895348 |
Oct 24, 2013 |
|
|
|
Current U.S.
Class: |
1/1 |
Current CPC
Class: |
E21B 7/04 20130101; E21B
47/0228 20200501; E21B 47/092 20200501; E21B 43/2406 20130101; G01V
3/26 20130101 |
International
Class: |
E21B 47/09 20060101
E21B047/09; E21B 47/022 20060101 E21B047/022 |
Claims
1. A method for determining a distance from a drilling well to a
magnetized target well, the method comprising: (a) deploying a
drill string in the drilling well, the drill string including at
least one magnetic field sensor in sensory range of magnetic flux
emanating from the magnetized target well; (b) making a plurality
of spaced apart magnetic field measurements in the drilling well;
(c) processing the spaced apart magnetic field measurements to
obtain a ratio of a magnetic field intensity to a first spatial
derivative of a magnetic field; and (d) processing the ratio
computed in (c) to obtain the distance from the drilling well to
the magnetized target well.
2. The method of claim 1, wherein the target well is magnetized
such that it includes a substantially periodic pattern of opposing
north-north (NN) magnetic poles and opposing south-south (SS)
magnetic poles spaced apart along a longitudinal axis thereof.
3. The method of claim 2, wherein the plurality of spaced apart
magnetic field measurements are made in (b) at locations adjacent
to one of the opposing NN or SS magnetic poles.
4. The method of claim 1, wherein: the magnetic field measurements
made in (b) are radially spaced apart; and the magnetic field
measurements are processed in (c) to obtain a ratio of the magnetic
field intensity of a radial component of the magnetic field to the
first spatial derivative of the radial component of the magnetic
field in the radial direction.
5. The method of claim 1, wherein: the magnetic field measurements
made in (b) are axially spaced apart; and the magnetic field
measurements are processed in (c) to obtain a ratio of the magnetic
field intensity of a radial component the magnetic field to the
first spatial derivative of an axial component of the magnetic
field in the axial direction.
6. The method of claim 1, further comprising: (e) processing the
magnetic field measurements made in (b) to compute a tool face to
target direction.
7. A method for determining a distance from a drilling well to a
magnetized target well, the method comprising: (a) deploying a
drill string in the drilling well, the drill string including a
magnetic field sensor in sensory range of magnetic flux emanating
from the magnetized target well; (b) making a plurality of spaced
apart magnetic field measurements in the drilling well; (c)
processing the spaced apart magnetic field measurements to obtain a
ratio of a magnetic field intensity to a second spatial derivative
of a magnetic field; and (d) processing the ratio computed in (c)
to obtain the distance from the drilling well to the magnetized
target well.
8. The method of claim 7, wherein the target well is magnetized
such that it includes a substantially periodic pattern of opposing
north-north (NN) magnetic poles and opposing south-south (SS)
magnetic poles spaced apart along a longitudinal axis thereof.
9. The method of claim 8, wherein the plurality of spaced apart
magnetic field measurements are made in (b) at locations adjacent
to one of the opposing NN or SS magnetic poles.
10. The method of claim 17, wherein: the magnetic field
measurements made in (b) are radially spaced apart; and the
magnetic field measurements are processed in (c) to obtain a ratio
of the magnetic field intensity of a radial component the magnetic
field to the second spatial derivative of the radial component of
the magnetic field in the radial direction.
11. The method of claim 10, wherein: the magnetic field
measurements made in (b) are axial spaced apart; the magnetic field
measurements are processed in (c) to obtain a ratio of the magnetic
field intensity of a radial component the magnetic field to the
second spatial derivative of the radial component of the magnetic
field in the axial direction.
12. The method of claim 10, further comprising: (e) processing the
magnetic field measurements made in (b) to compute a tool face to
target direction.
13. A method for determining a distance from a drilling well to a
magnetized target well, the method comprising: (a) deploying a
drill string in the drilling well, the drill string including a
magnetic field sensor in sensory range of magnetic flux emanating
from the magnetized target well; (b) making a plurality of spaced
apart magnetic field measurements in the drilling well; (c)
processing the spaced apart magnetic field measurements to obtain a
ratio of a first spatial derivative of the magnetic field and a
second spatial derivative of the magnetic field; and (d) processing
the ratio computed in (c) to obtain the distance from the drilling
well to the magnetized target well.
14. The method of claim 13, wherein the target well is magnetized
such that it includes a substantially periodic pattern of opposing
north-north (NN) magnetic poles and opposing south-south (SS)
magnetic poles spaced apart along a longitudinal axis thereof.
15. The method of claim 14, wherein the plurality of spaced apart
magnetic field measurements are made in (b) at locations adjacent
to one of the opposing NN or SS magnetic poles.
16. The method of claim 13, wherein: the magnetic field
measurements made in (b) are radially spaced; and the magnetic
field measurements are processed in (c) to obtain a ratio of the
first spatial derivative of a radial component of the magnetic
field in the radial direction to the second spatial derivative of
the radial component of the magnetic field in the radial
direction.
17. The method of claim 13, wherein: the magnetic field
measurements made in (b) are axially spaced; and the magnetic field
measurements are processed in (c) to obtain a ratio of the first
spatial derivative of an axial component of the magnetic field in
the axial direction to the second spatial derivative of a radial
component of the magnetic field in the axial direction.
18. The method of claim 13, wherein: the magnetic field
measurements made in (b) are both axially spaced and radially
spaced; the magnetic field measurements are processed in (c) to
obtain a ratio of the first spatial derivative of an axial
component of the magnetic field in the axial direction to the
second spatial derivative of a radial component of the magnetic
field in the radial direction.
19. The method of claim 13, wherein: the magnetic field
measurements made in (b) are both axially spaced and radially
spaced; the magnetic field measurements are processed in (c) to
obtain a ratio of the first spatial derivative of a radial
component of the magnetic field in the radial direction to the
second spatial derivative of the radial component of the magnetic
field in the axial direction.
20. The method of claim 13, further comprising: (e) processing the
magnetic field measurements made in (b) to compute a tool face to
target direction.
Description
CROSS REFERENCE TO RELATED APPLICATIONS
[0001] This application claims priority to and the benefit of U.S.
Provisional Patent Application No. 61/894,460, filed 24 Oct. 2013,
which is incorporated by reference herein.
FIELD OF THE INVENTION
[0002] Disclosed embodiments relate generally to drilling and
surveying subterranean boreholes such as for use in oil and natural
gas exploration and more particularly to methods for determining a
distance between a twin well and a magnetized target well using
first spatial derivatives and second spatial derivatives of a
measured magnetic field.
BACKGROUND INFORMATION
[0003] Magnetic ranging measurements may be used to obtain a
distance and a direction to an adjacent well. For example, commonly
assigned U.S. Pat. No. 7,656,161 discloses a technique in which a
predetermined magnetic pattern is deliberately imparted to a
plurality of casing tubulars. These tubulars, thus magnetized, are
coupled together and lowered into the adjacent well (the target
well) to form a magnetized section of casing string typically
including a plurality of longitudinally spaced pairs of opposing
magnetic poles. Measurements of the magnetic field may then be
utilized to survey and guide drilling of a drilling well (e.g. a
twin well) relative to the target well. The distance between the
twin and target wells may be determined from various magnetic field
measurements made in the twin well (as further disclosed in
commonly assigned U.S. Pat. No. 7,617,049). These well twinning
techniques may be advantageously utilized, for example, in steam
assisted gravity drainage (SAGD) applications in which horizontal
twin wells are drilled to recover heavy oil from tar sands.
[0004] While the above described methodology has been successfully
utilized in well twinning applications, there is room for yet
further improvement. For example, it can be difficult to accurately
remove the earth's magnetic field from the measured magnetic field
since the attitude of the drilling well is not generally known with
precision. Moreover, since the distance between the two wells is
obtained from the measured magnetic field strength (intensity), any
changes in the strength of the casing magnetization may cause a
corresponding error in the obtained distance (e.g., a decay in the
casing magnetization may cause the distance to be underestimated).
Therefore there is a need for improved ranging methodologies.
SUMMARY
[0005] Methods for determining a distance from a drilling well to a
magnetized target well are disclosed. The methods include acquiring
magnetic field measurements from the drilling well. A drill string
is deployed in the drilling well and includes at least one magnetic
field sensor in sensory range of magnetic flux emanating from the
magnetized target well. The acquired magnetic field measurements
are made at a plurality of spaced apart locations, e.g., at a
plurality of spaced apart axial and/or radial locations in the
drilling well. The acquired magnetic field measurements are
processed to obtain a ratio including at least one of the
following: (i) a ratio of a magnetic field intensity to a first
spatial derivative of a magnetic field, (ii) a ratio of a magnetic
field intensity to a second spatial derivative of a magnetic field,
and (iii) a ratio of a first spatial derivative of a magnetic field
to a second spatial derivative of the magnetic field. The ratio (or
ratios) are then processed to obtain the distance from the drilling
well to the magnetized target well.
[0006] The disclosed embodiments may provide various technical
advantages. For example, the disclosed methods may improve the
accuracy of the distances determined via magnetic ranging by
reducing the dependence of the magnetic ranging measurements on the
strength of the target magnetization. Moreover, certain of the
disclosed embodiments may obviate the need to remove the earth's
magnetic field from the measured magnetic field.
[0007] This summary is provided to introduce a selection of
concepts that are further described below in the detailed
description. This summary is not intended to identify key or
essential features of the claimed subject matter, nor is it
intended to be used as an aid in limiting the scope of the claimed
subject matter.
BRIEF DESCRIPTION OF THE DRAWINGS
[0008] For a more complete understanding of the disclosed subject
matter, and advantages thereof, reference is now made to the
following descriptions taken in conjunction with the accompanying
drawings, in which:
[0009] FIG. 1 depicts a prior art arrangement for a SAGD well
twinning operation.
[0010] FIG. 2 depicts a prior art magnetization of a wellbore
tubular.
[0011] FIG. 3 depicts a flow chart of one example of a disclosed
method embodiment for determining a distance between a drilling
well and a magnetized target well.
[0012] FIG. 4 depicts a plot of the magnetic field about a
magnetized casing string.
[0013] FIGS. 5A and 5B depict plots of the axial and radial
components (B.sub.z and B.sub.r) of the magnetic field as a
function of normalized axial position along the target well at
various distances from the target well.
[0014] FIGS. 6A, 6B, and 6C depict plots of the three independent
first spatial derivatives of the magnetic field as a function of
normalized axial position along the target well at various
distances from the target well.
[0015] FIGS. 7A, 7B, 7C, and 7D depict plots of the four
independent second spatial derivatives of the magnetic field as a
function of normalized axial position along the target well at
various distances from the target well.
[0016] FIGS. 8A and 8B depict plots of various ratios of a magnetic
field intensity to a first spatial derivative of the magnetic field
as a function of the actual distance to the magnetized target.
[0017] FIGS. 9A and 9B depict plots of various ratios of a magnetic
field intensity to a second spatial derivative of the magnetic
field as a function of the actual distance to the magnetized
target.
[0018] FIGS. 10A, 10B, 10C, and 10D depict plots of various ratios
of a first spatial derivative of a magnetic field to a second
spatial derivative of the magnetic field as a function of the
actual distance to the magnetized target.
DETAILED DESCRIPTION
[0019] FIG. 1 schematically depicts one example of a well twinning
application such as a SAGD twinning operation. Common SAGD twinning
operations require a horizontal twin well 20 to be drilled a
substantially fixed distance substantially directly above a
horizontal portion of the target well 30 (e.g., not deviating more
than about 1-2 meters up or down or to the left or right of the
lower well). In the exemplary embodiment shown, the lower (target)
well 30 is drilled first, for example, using conventional
directional drilling and MWD techniques. However, the disclosed
embodiments are not limited in regard to which of the wells is
drilled first. The target wellbore 30 is then cased using a
plurality of premagnetized tubulars (such as those shown on FIG. 2
described below) to form a magnetized casing string 35. In the
embodiment shown, drill string 24 includes at least one tri-axial
magnetic field measurement sensor 28 deployed in close proximity to
the drill bit 22. Sensor 28 is used to passively measure the
magnetic field about target well 30 as the twin well is drilled.
Such passive magnetic field measurements are then utilized to guide
continued drilling of the twin well 20 along a predetermined path
relative to the target well 30 (e.g., as described in U.S. Pat.
Nos. 7,617,049, 7,656,161, and 8,026,722, each of which is fully
incorporated by reference herein).
[0020] With reference now to FIG. 2, one example tubular 60
magnetized as described in the '722 patent is shown. The depicted
tubular 60 embodiment includes a plurality of discrete magnetized
zones 62 (typically three or more). Each magnetized zone 62 may be
thought of as a discrete cylindrical magnet having a north N pole
on one longitudinal end thereof and a south S pole on an opposing
longitudinal end thereof such that a longitudinal magnetic flux 68
is imparted to the tubular 60. Tubular 60 further includes a single
pair of opposing north-north NN poles 65 at the midpoint thereof.
The purpose of the opposing magnetic poles 65 is to focus magnetic
flux outward from tubular 60 as shown at 70 (or inward for opposing
south-south poles as shown at 72). The tubulars may be magnetized,
for example, using the apparatus disclosed in U.S. Pat. No.
7,538,650, which is fully incorporated by reference herein.
[0021] With continued reference to FIG. 1, the casing string 35 is
formed by joining (threadably connecting) premagnetized tubulars in
the target well 30. In one embodiment, the resultant string 35 has
a single pair of opposing magnetic poles in the central region (the
middle third) of each tubular. Thus the pairs of opposing magnetic
poles (NN or SS) are spaced at intervals about equal to the length
of tubulars, while the period of the magnetic field pattern (e.g.,
the distance from one a NN pair of opposing magnetic poles to the
next NN pair) is about twice the length of a tubular.
[0022] As described above, drill string 20 may include a triaxial
magnetic field sensor 28. The depicted embodiment of the sensor 28
includes three mutually orthogonal magnetic field sensors, one of
which is oriented substantially parallel with the borehole axis
(M.sub.Z). Sensor 28 may thus be considered as determining a plane
(defined by M.sub.X and M.sub.Y) orthogonal to the borehole axis
and a pole (M.sub.Z) parallel to the borehole axis of the twin
well, where M.sub.X, M.sub.Y, and M.sub.Z represent measured
magnetic field vectors in the x, y, and z directions.
[0023] The magnetic field about the magnetized casing string may be
measured and represented, for example, as a vector whose
orientation depends on the location of the measurement point within
the magnetic field. In order to determine the magnetic field vector
due to the target well (e.g., target well 30) at any point
downhole, the magnetic field of the earth may be subtracted from
the measured magnetic field vector using means known to those of
ordinary skill in the art. The magnetic field of the earth
(including both magnitude and direction components) may be known,
for example, from previous geological survey data or a geomagnetic
model. It will be understood that in certain embodiments such
subtraction of the magnetic field of the earth is not required.
[0024] It will be appreciated that the disclosed embodiments are
not limited to the depictions of FIGS. 1 and 2. For example, the
disclosure is not limited to SAGD applications. Rather, exemplary
methods in accordance with this disclosure may be utilized to drill
twin wells having substantially any relative orientation for
substantially any application. Moreover, the disclosure is not
limited to any particular magnetization pattern or spacing of pairs
of opposing magnetic poles on the target well.
[0025] FIG. 3 depicts a flow chart of one example of a disclosed
method embodiment 100 for determining a distance between a drilling
well and a magnetized target well (e.g., as depicted on FIG. 1).
The method includes acquiring a plurality of axially and or
radially spaced magnetic field measurements at 110. The magnetic
field measurements may then be processed at 120 to compute first
spatial derivatives and second spatial derivatives of the magnetic
field. The first spatial derivatives and second spatial derivatives
may be further processed at 130 to compute one or more of the
following ratios: (i) a ratio of the magnetic field intensity to a
first spatial derivative of the magnetic field, (ii) a ratio of the
magnetic field intensity to a second spatial derivative of the
magnetic field, and/or (iii) a ratio of a first spatial derivative
of the magnetic field to a second spatial derivative of the
magnetic field. The computed ratio or ratios may then be further
processed to obtain the distance between the drilling well and the
magnetized target well at 140.
[0026] The plurality of axial and/or radially spaced magnetic field
measurements may be acquired at 110 using magnetic field sensors
deployed in a drill string in the drilling well (e.g., sensor 28
deployed in drill string 24 in drilling well 20 in FIG. 1). In
certain embodiments, the spaced magnetic field measurements may be
made using a single triaxial magnetic field sensor. For example,
axially spaced measurements may be obtained via moving the drill
string axially in the wellbore (in the uphole or downhole
direction) between measurements. Radially spaced measurements may
be obtained by rotating an off-centered (eccentered) sensor to
various toolface angles between measurements. In other embodiments,
the drill string may include a plurality of axially and/or radially
spaced magnetic field sensors. For example, two, three, or more
axially spaced measurements may be acquired via corresponding
magnetic field sensors deployed in the drill string (e.g., at half
meter intervals along the length of the string). Radially spaced
measurements may be acquired via corresponding magnetic field
sensors deployed about the circumference of the drill string (e.g.,
first and second diametrically opposed sensors or three or more
sensors deployed at suitable angular intervals about the
circumference). Radially spaced measurements may also be acquired
using corresponding sensors having different degrees of
eccentricity (e.g., a central sensor and one or more eccentered
sensors). The magnetic field sensors may also be offset both
axially and radially (e.g., first and second axially spaced sensors
having one or more eccentered sensors located axially between
them). The disclosed method embodiments are not limited to any
particular magnetic field sensor configuration and/or spacing.
[0027] The magnetic field measurements may be resolved into three
orthogonal components which can in turn be defined, for example, as
highside, lateral, and along-hole or axial directions (or x, y, and
z directions as described above). The highside and lateral
components may also be resolved into polar coordinates, designated,
for example, by a radial intensity and a toolface-to-target
direction. Four magnetic field gradients (first spatial derivatives
of the magnetic field) may be defined based on the axial and radial
components. However, since the magnetic field is magnetostatic and
current-free, its curl is zero and only three of these gradients
are independent as indicated below:
.differential. B r .differential. r ; .differential. B z
.differential. z ; and .differential. B r .differential. z =
.differential. B z .differential. r ( 1 ) ##EQU00001##
[0028] where B.sub.z and B.sub.r represent the intensity of the
measured magnetic field in the axial (z) and radial (r) directions.
Four independent second spatial derivatives of the magnetic field
may also be obtained based on the axial and radial components of
the magnetic field. They are as follows:
.differential. 2 B r .differential. r 2 ; .differential. 2 B z
.differential. z ; .differential. 2 B r .differential. z 2 =
.differential. 2 B z .differential. r .differential. z ; and
.differential. 2 B z .differential. r 2 = .differential. 2 B r
.differential. r .differential. z ( 2 ) ##EQU00002##
[0029] It will be understood that at least two spaced apart
magnetic field measurements are generally required to obtain a
first spatial derivative of the magnetic field (a gradient of the
magnetic field) and that at least three spaced apart magnetic field
measurements are generally required to obtain a second spatial
derivative of the magnetic field (a curvature of the magnetic
field).
[0030] The magnetic field gradients may be computed at 120, for
example, from first and second spaced apart magnetic field
measurements. For example, the gradient of the axial component of
the magnetic field in the axial direction
(.differential.B.sub.z/.differential.z) may be obtained as
follows:
.differential. B z .differential. z = .DELTA. B z .DELTA. z ( 3 )
##EQU00003##
[0031] where .DELTA.B.sub.z represents the difference in the axial
component of the magnetic field between the first and second
measurement positions (i.e., .DELTA.B.sub.z=B.sub.z2-B.sub.z1) and
.DELTA.z represents the axial measurement spacing (the distance
between the first and second measurement positions, i.e.,
.DELTA.z=z.sub.2-z.sub.1). Gradients of the radial component of the
magnetic field and/or in the radial direction may be similarly
computed.
[0032] The second spatial derivatives may be computed at 120, for
example, from first, second, and third spaced apart magnetic field
measurements. For example, the curvature of the axial component of
the magnetic field in the axial direction (.differential..sup.2
B.sub.z/.differential.z.sup.2) may be obtained as follows:
.differential. 2 B z .differential. z 2 = ( .DELTA. B z .DELTA. z (
2 ) - .DELTA. B z .DELTA. z ( 1 ) ) .DELTA. z = B z 3 - 2 B z 2 + B
z 1 ( .DELTA. z ) 2 ( 4 ) ##EQU00004##
[0033] where
.DELTA. B z .DELTA. z ( 1 ) ##EQU00005##
represents the magnetic field gradient between the first and second
axial positions,
.DELTA. B z .DELTA. z ( 2 ) ##EQU00006##
represents the magnetic field gradient between the second and third
axial positions, and .DELTA.z represents the axial measurement
spacing. The second spatial derivatives may also be obtained, for
example, by fitting three or more spaced measurements to a function
such as a polynomial and then differentiating the function. Second
spatial derivatives of the radial component of the magnetic field
and/or in the radial direction may be similarly computed.
[0034] Owing to the dimensional constraints on downhole tools, the
radial measurement spacing tends to be limited to about 0.1 meters
or less. The spacing in the axial direction is not physically
constrained in the same way; however, it may be advantageous for
the axial measurement spacing to be less than about a few meters in
order to maintain good resolution and to avoid complications caused
by tool curvature. The short radial measurement spacing tends to
increases sensitivity to noise such that in certain operations it
may be advantageous to use the axially distributed measurements
.differential.B.sub.r/.differential.z,
.differential.B.sub.z/.differential.z,
.differential..sup.2B.sub.r/.differential.z.sup.2, and
.differential..sup.2B.sub.z/.differential.z.sup.2 when
possible.
[0035] Variations in the first spatial derivatives and the second
spatial derivatives of the magnetic field with position relative to
a magnetized target well may be evaluated using a magnetic model.
For example, a magnetized casing string having a repeating magnetic
pattern along the axis of the string (e.g., as described above with
respect to FIGS. 1 and 2) may be modelled as a repeating series of
point sources (monopoles) and/or line sources distributed along the
centerline of the string. For a monopole model, the field at any
point (r, z) from a point source located at (0, zp) may expressed
as follows:
B z = P 4 .pi. ( z - zp ) [ ( z - zp ) 2 + r 2 ] 1.5 ( 5 ) B r = P
4 .pi. r [ ( z - zp ) 2 + r 2 ] 1.5 ( 6 ) ##EQU00007##
[0036] where P represents the strength of each of the magnetic
poles and 0.ltoreq.p<1 and represents the axial location along
the repeating magnetic pattern (where the positions p=0, 1, . . .
are adjacent NN opposing magnetic poles). For a line source model,
the field at any point (r, z) from a line source of length L
centered at (0, zp) may expressed as follows:
B z = P 4 .pi. L [ 1 ( z - zp - L / 2 ) 2 + r 2 - 1 ( z - zp + L /
2 ) 2 + r 2 ] ( 7 ) B r = P 4 .pi. Lr [ z - zp + L / 2 ( z - zp + L
/ 2 ) 2 + r 2 - z - zp - L / 2 ( z - zp - L / 2 ) 2 + r 2 ] ( 8 )
##EQU00008##
[0037] FIG. 4 depicts a plot of the actual magnetic field about a
magnetized casing string. The field is represented as a plot of the
axial component of the magnetic field versus the radial component
of the magnetic field. The magnetic field is further plotted at
various radial distances from the string. The casing string was
magnetized with a repeating pattern of opposing magnetic poles such
that the pattern repeats with a period of twice the length of the
tubulars that make up the string (as described above). It may be
noted that the casing magnetization in this example is mildly
asymmetric with the left side of the plot being larger than the
right side, possibly indicating that joints magnetized with one
polarity retained slightly more magnetization than the others (the
disclosed embodiments are of course not limited in this regard).
This fact will aid in determining the sensitivity of a ranging
technique to the absolute magnetization of the target as ideally
the calculated distance should be the same for both joints.
[0038] When ranging to a target well magnetized as described above,
drilling may be stopped and magnetic surveys taken at locations
corresponding to maximum radial flux from the target (i.e. adjacent
the NN or SS opposing magnetic poles located at the approximate
midpoint of each tubular). At these locations the axial field from
the target tends to be small (near zero) while the radial field
tends to be at a maximum. These locations correspond to the left
and right sides of the plot depicted on FIG. 4. The gradients
.differential.B.sub.z/.differential.z and
.differential.B.sub.r/.differential.r are relatively large at these
locations while .differential.B.sub.r/.differential.z is small
(near zero). Of the second spatial derivatives,
.differential..sup.2B.sub.r/.differential.r.sup.2 and
.differential..sup.2B.sub.r/.differential.z.sup.2 tend to be large,
while .differential..sup.2B.sub.z/.differential.r.sup.2 and
.differential..sup.2B.sub.z/.differential.z.sup.2 are small (near
zero). Since measurements of small quantities tend to be
susceptible to noise, it may be advantageous make use of the larger
values .differential.B.sub.r/.differential.r,
.differential.B.sub.z/.differential.z,
.differential..sup.2B.sub.r/.differential.r.sup.2, and
.differential..sup.2B.sub.r/.differential.z.sup.2, and particularly
the long baseline measurements
.differential.B.sub.z/.differential.z and
.differential..sup.2B.sub.r/.differential.z.sup.2.
[0039] FIGS. 5A and 5B depict plots of the axial and radial
components (B.sub.z and B.sub.r) of the magnetic field as a
function of normalized axial position along the target well at
various distances from the target well. The joint ends are located
at normalized axial positions of 1.0 and 2.0 while the opposing
magnetic poles are located at normalized axial positions of 0.5,
1.5, and 2.5 (with SS opposing magnetic poles being located at 0.5
and 2.5 and a NN opposing magnetic pole being located at 1.5).
Consistent with the plot depicted on FIG. 4, the radial component
has maxima at axial positions of 0.5, 1.5, and 2.5 (adjacent to the
opposing magnetic poles).
[0040] FIGS. 6A, 6B, and 6C depict plots of the three independent
magnetic field gradients (first spatial derivatives) as a function
of normalized axial position along the target well at various
distances from the target well. FIG. 6A depicts the gradient of the
intensity of the radial magnetic field component in the radial
direction .differential.B.sub.r/.differential.r. FIG. 6B depicts
the gradient of the intensity of the axial magnetic field component
in the axial direction .differential.B.sub.z/.differential.z. And
FIG. 6C depicts the gradient of the intensity of the radial
magnetic field component in the axial direction
.differential.B.sub.r/.differential.z (which is equal to the
gradient of the intensity of the axial magnetic field component in
the radial direction .differential.B.sub.z/.differential.r). FIGS.
6A and 6B show that .differential.B.sub.r/.differential.r and
.differential.B.sub.z/.differential.z have maxima at axial
positions of 0.5, 1.5, and 2.5 (adjacent the opposing magnetic
poles). FIG. 6C shows that .differential.B.sub.r/.differential.z is
approximately zero at the same axial positions.
[0041] FIGS. 7A, 7B, 7C, and 7D depict plots of the four
independent second spatial derivatives of the magnetic field as a
function of normalized axial position along the target well at
various distances from the target well. FIG. 7A depicts the second
spatial derivative of the radial component of the magnetic field in
the radial direction
.differential..sup.2B.sub.r/.differential.r.sup.2. FIG. 7B depicts
the second spatial derivative of the radial component of the
magnetic field in the axial direction
.differential..sup.2B.sub.r/.differential.z.sup.2. FIG. 7C depicts
the second spatial derivative of the axial component of the
magnetic field in the radial direction
.differential..sup.2B.sub.z/.differential.r.sup.2. FIG. 7D depicts
the second spatial derivative of the axial component of the
magnetic field in the axial direction
.differential..sup.2B.sub.z/.differential.z.sup.2. FIGS. 7A and 7B
show that .differential..sup.2B.sub.r/.differential.r.sup.2 and
.differential..sup.2B.sub.r/.differential.z.sup.2 have maxima at
axial positions of 0.5, 1.5, and 2.5 (adjacent the opposing
magnetic poles). FIGS. 7C and 7D show that
.differential..sup.2B.sub.z/.differential.r.sup.2 and
.differential..sup.2B.sub.z/.differential.z.sup.2 are approximately
zero at the same axial positions.
[0042] When the magnetic field measurements are made at axial
positions adjacent (or nearly adjacent) to the opposing magnetic
poles, the magnetic field intensity, the first spatial derivatives,
and the second spatial derivatives may be approximated, for
example, from Equations 5 and 6 above (the monopole approximation).
Thus, for example, when z=zp the magnetic field intensities may be
expressed as follows:
B z .apprxeq. 0 ( 9 ) B r .apprxeq. P 4 .pi. r 2 ( 10 )
##EQU00009##
[0043] The first spatial derivatives may be also be expressed, for
example as follows:
.differential. B z .differential. z .apprxeq. P 4 .pi. r 3 ( 11 )
.differential. B r .differential. r .apprxeq. P 4 .pi. r 3 ( 12 )
.differential. B r .differential. z .apprxeq. .differential. B z
.differential. r .apprxeq. 0 ( 13 ) ##EQU00010##
[0044] The second spatial derivatives may also be expressed, for
example, as follows:
.differential. 2 B r .differential. r 2 .apprxeq. 3 P 2 .pi. r 4 (
14 ) .differential. 2 B r .differential. z 2 .apprxeq. 3 P 4 .pi. r
4 ( 15 ) .differential. 2 B z .differential. r 2 .apprxeq.
.differential. 2 B r .differential. z 2 .apprxeq. 0 ( 16 )
##EQU00011##
[0045] As described above, the intent of the magnetic ranging
measurements is to determine the relative position of the drilling
well with respect to the magnetized target well, for example, via
determining a distance and direction from the drilling well to the
target well. The toolface direction (the direction in the plane
normal to the tool axis) towards the target may be obtained from a
ratio of the two components measured in that plane (e.g., a ratio
of the x and y components of the measured magnetic field). The
distance to the target may be found from a ratio of a magnetic
field intensity to a first spatial derivative of the magnetic
field, a ratio of a magnetic field intensity to a second spatial
derivative of the magnetic field, and/or a ratio of a first spatial
derivative of the magnetic field to a second spatial derivative of
the magnetic field. The use of one or more of the following ratios
may be advantageous in that the ratios are independent of the
strength of the magnetic poles. The use of multiple ratios may
further improve the accuracy of the obtained distance by giving
corresponding multiple independent measurements.
[0046] When the magnetic field measurements are made at axial
positions adjacent (or nearly adjacent) to the opposing magnetic
poles, the ratios may be approximated from certain of Equations 9
through 16 above. The distance to the target may be expressed in
terms of example ratios of a magnetic field intensity to a first
spatial derivative of the magnetic field, for example, as
follows:
r .apprxeq. B r .differential. B z .differential. z ( 17 ) r
.apprxeq. - 2 B r .differential. B r .differential. r ( 18 )
##EQU00012##
[0047] The distance to the target may also be expressed in terms of
example ratios of a magnetic field intensity to a second spatial
derivative of the magnetic field, for example, as follows:
r .apprxeq. [ 6 B r .differential. 2 B r .differential. r 2 ] 1 2 (
19 ) r .apprxeq. [ - 3 B r .differential. 2 B r .differential. z 2
] 1 2 ( 20 ) ##EQU00013##
[0048] The distance to the target may be further expressed in terms
of example ratios of a first spatial derivative of the magnetic
field to a second spatial derivative of the magnetic field, for
example, as follows:
r .apprxeq. 6 .differential. B z .differential. z .differential. 2
B r .differential. r 2 ( 21 ) r .apprxeq. - 3 .differential. B z
.differential. z .differential. 2 B r .differential. z 2 ( 22 ) r
.apprxeq. - 3 .differential. B r .differential. r .differential. 2
B r .differential. r 2 ( 23 ) r .apprxeq. 1.5 .differential. B r
.differential. r .differential. 2 B r .differential. z 2 ( 24 )
##EQU00014##
[0049] The performance of these functions (equations 17 through 24)
may be estimated using the model of the magnetized target shown on
FIG. 4. A transform may be developed to convert the ratio to its
corresponding actual distance. The ratios between a magnetic field
intensity and a first spatial derivative of the magnetic field
(given in equations 17 and 18) are evaluated in the plots shown on
FIGS. 8A and 8B. FIG. 8A depicts a plot of the ratio
B r .differential. B z .differential. z ##EQU00015##
versus actual distance at axial positions of 0.5, 1.5, and 2.5. In
this example the ratio seems to be poorly suited to determining
distance as it is substantially independent of distance. FIG. 8B
depicts a plot of the ratio
- 2 B r .differential. B r .differential. r ##EQU00016##
versus actual distance at axial positions of 0.5, 1.5, and 2.5. In
this example, the ratio varies monotonically with distance. The
separation between the two curves at larger distances indicates
that the ratio may be somewhat sensitive to the absolute intensity
of the magnetic poles.
[0050] The ratios between a magnetic field intensity and a second
spatial derivative of the magnetic field (given in equations 19 and
20) are evaluated at normalized axial positions of 0.5, 1.5, and
2.5 in the plots shown on FIGS. 9A and 9B. FIG. 9A depicts a plot
of the ratio
( 6 B r .differential. 2 B r .differential. r 2 ) 1 2
##EQU00017##
versus actual distance while FIG. 9B depicts a plot of the
ratio
( - 3 B r .differential. 2 B r .differential. z 2 ) 1 2
##EQU00018##
versus actual distance. In these examples, the ratios vary
monotonically with distance and may therefore be suitable for use
in distance determination. The separation between the two curves in
each plot indicates that these ratios may be somewhat sensitive to
the absolute intensity of the magnetic poles.
[0051] The ratios between a first spatial derivative of the
magnetic field and a second spatial derivative of the magnetic
field (given in equations 21 through 24) are evaluated at
normalized axial positions of 0.5, 1.5, and 2.5 in the plots shown
on FIGS. 10A, 10B, 10C, and 10D. FIG. 10A depicts a plot of the
ratio
6 .differential. B z .differential. z .differential. 2 B r
.differential. r 2 ##EQU00019##
versus actual distance. In this example the ratio is a strong
monotonic function of the distance making it a good candidate for
distance determination. FIG. 10B depicts a plot of the ratio
- 3 .differential. B z .differential. z .differential. 2 B r
.differential. z 2 ##EQU00020##
versus actual distance. The second spatial derivative in this ratio
may also be determined by measuring .differential./.differential.z
(.differential.B.sub.z/.differential.r) or
.differential./.differential.r(.differential.B.sub.z/.differential.z).
FIG. 10C depicts a plot of the ratio
- 3 .differential. B r .differential. r .differential. 2 B r
.differential. r 2 ##EQU00021##
versus actual distance. In these examples, the ratios vary
monotonically with distance and may therefore be suitable for use
in distance determination. The separation between the two curves in
FIGS. 10A and 10B indicates that these ratios may be somewhat
sensitive to the absolute intensity of the magnetic poles. The
ratio in FIG. 10C shows very little sensitivity to the absolute
intensity of the magnetic poles. FIG. 10D depicts a plot of the
ratio
1.5 .differential. B r .differential. r .differential. 2 B r
.differential. z 2 ##EQU00022##
versus actual distance. The second spatial derivative in this ratio
may also be determined by measuring .differential./.differential.r
(.differential.B.sub.z/.differential.z) or
.differential./.differential.z
(.differential.B.sub.z/.differential.r). In this example, the ratio
is not well correlated with distance.
[0052] It will be understood that method 100 may be performed using
uphole and/or downhole processors. The disclosed embodiments are
not limited in this regard. For example, magnetic field
measurements may be transmitted to the surface (using any suitable
telemetry techniques). The distance may then be computed at the
surface and further used to compute a new drilling direction which
may then be transmitted back to the tool. Alternatively, the
magnetic field measurements may be processed downhole to obtain the
distance, for example, using one or more look up tables to
correlate the computed ratio(s) to distance. The obtained distance
may then be used to compute a new drilling direction downhole which
may be implemented as part of a closed loop well twinning
methodology.
[0053] While the aforementioned examples make use of a target well
is magnetization having axially spaced opposing magnetic poles it
will be understood that the disclosed embodiments are not so
limited. The use of first spatial derivatives and second spatial
derivatives of the magnetic field and ratios including those
derivatives may be used with substantially any suitable target well
magnetization.
[0054] Although a method for magnetic gradient and curvature based
ranging and certain advantages thereof have been described in
detail, it should be understood that various changes, substitutions
and alternations can be made herein without departing from the
spirit and scope of the disclosure as defined by the appended
claims.
* * * * *