Note : Les descriptions sont présentées dans la langue officielle dans laquelle elles ont été soumises.
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MAGNETIC GRADIENT AND CURVATURE BASED RANGING METHOD
CROSS REFERENCE TO RELATED APPLICATIONS
[0001] This application claims priority to and the benefit of US Provisional
Patent
Application No. 61/894,460, filed 24 October 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. Patent 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
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determined from various magnetic field measurements made in the twin well (as
further
disclosed in commonly assigned U.S. Patent 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
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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.
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[0 0 1 1] 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 and
Br) 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.
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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.
Patents 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
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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. Patent 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 (Mz). Sensor 28
may thus be considered as determining a plane (defined by Mx and My)
orthogonal to the
borehole axis and a pole (Mz) parallel to the borehole axis of the twin well,
where Mx, My, and
Mz 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
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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
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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
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(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:
aBr aBz , aBr aBz
¨;¨; an a ¨ = = (1)
ar az az ar
[0028] where Bz and Br 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:
a2Br.a2Bz.a2Br a2Bz. a2Bz a2Br
2
ar2 ' az2 ' az2 araz ' and ar2 ¨ar.az ( )
[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
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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 (aBz/az) may be
obtained as follows:
aBz = ABz
(3)
az Az
[0031] where ABz represents the difference in the axial component of the
magnetic field
between the first and second measurement positions (i.e., ABz = Bz2 Bz1) and
Az represents
the axial measurement spacing (the distance between the first and second
measurement
positions, i.e., Az = z2 ¨ z1). 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 (a 2 Bz/az2)
may be obtained
as follows:
(ABz, ABz(i ))
= (
a2Bz Bz3-2Bz2+Bzi 4)
3z2 Az (Az)2
[0033] where ¨ABz (1) represents the magnetic field gradient between the first
and second
Az
axial positions, ¨AABz (2) represents the magnetic field gradient between the
second and third
axial positions, and Az represents the axial measurement spacing. The second
spatial
derivatives may also be obtained, for example, by fitting three or more spaced
measurements
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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 aBr/az, aBz/az, a2Br/az2, and
a2Bz/az2 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:
P
B == (z-zp)
(5)
Z 47 [(z_zo2 +r2ii=5
P r
13r = (6)
47 [(z_zo2 +r2ii 5
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[0036] where P represents the strength of each of the magnetic poles and 0 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:
1 1
= ________________________ ,
Bz = 47L [v(z-zp-L/2)2+r2 -1(z-zp+L/2)2+r2 (7)
B = ¨ = z-zp+L/2 z-zp-L/2 (8)
r
' 47Lr , (z-zp+L/2)2+r2 Al(z-zp-L/2)2+r2
[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
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(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 a/3,/az and aBriar
are relatively
large at these locations while aBriaz is small (near zero). Of the second
spatial derivatives,
a2/3,/ar2 and a2/3,/az2 tend to be large, while a2 13,/ar2 and a2/3,/az2 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 aBriar, a/3,/az, a2/3,/ar2, and
a2/3,/az2, and
particularly the long baseline measurements a 13,/az and a2 13,/az2.
[0039] FIGS. 5A and 5B depict plots of the axial and radial components (B, and
13,) 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 a/3Jan FIG. 6B depicts
the gradient
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of the intensity of the axial magnetic field component in the axial direction
aBz/az. And FIG.
6C depicts the gradient of the intensity of the radial magnetic field
component in the axial
direction aBriaz (which is equal to the gradient of the intensity of the axial
magnetic field
component in the radial direction aBz/ar). FIGS. 6A and 6B show that aBriar
and aBz/az
have maxima at axial positions of 0.5, 1.5, and 2.5 (adjacent the opposing
magnetic poles).
FIG. 6C shows that aBriaz 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 a2Briar2.
FIG. 7B depicts
the second spatial derivative of the radial component of the magnetic field in
the axial direction
a2Briaz2. FIG. 7C depicts the second spatial derivative of the axial component
of the
magnetic field in the radial direction a2Bziar2. FIG. 7D depicts the second
spatial derivative
of the axial component of the magnetic field in the axial direction a2Bziaz2.
FIGS. 7A and
7B show that a 2Brlar2 and a2BrIaz2 have maxima at axial positions of 0.5,
1.5, and 2.5
(adjacent the opposing magnetic poles). FIGS. 7C and 7D show that a2Bziar2 and
a 2BziaZ2
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
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Equations 5 and 6 above (the monopole approximation). Thus, for example, when
z = zp the
magnetic field intensities may be expressed as follows:
Bz P=-=-= 0 (9)
P
Br P.-- 47r2
(10)
[0043] The first spatial derivatives may be also be expressed, for example as
follows:
aBz P
(11)
az 47r3
aBr P
(12)
ar 27r3
aBr aBz
¨az = ¨ar(13)
[0044] The second spatial derivatives may also be expressed, for example, as
follows:
(14)
ar2 ¨= 2= 7r4
(15)
az2 ¨= 4= 7r4
a2Bz a2Br 0
(16)
ar2 "..' az2
[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
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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:
Br
r -,-L-: __________________________________________________________ (17)
3B/3z
Br
(18)
r 2 aBr/ar
[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:
B 11/2
r P=-=-= [62Brriar2 (19)
a _____________________________________
Br 1 (20)1/2
r '''''' [ 3 a2Baz2
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[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 6
aBz/iar2 az
(21)
' a2Br
aBz/az
r
(22)
'''' 3 a2B,/az2
aBr/ar
r ' 3 a2Briar2 (23)
aBr/ar
r '''' 1.5 a2B,/az2 (24)
[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
Br
evaluated in the plots shown on FIGS. 8A and 8B. FIG. 8A depicts a plot of the
ratio
aBz/az
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
Br
depicts a plot of the ratio 2 aBriar versus actual distance at 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.
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[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 a2Briar2) Br \ 1/2
versus actual distance while FIG. 9B depicts a plot of the ratio
(_3 a2B,/az2Br )1/2
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.
aBz/az
10A depicts a plot of the ratio 6 a2Briar2 versus actual distance. In this
example the ratio is a
strong monotonic function of the distance making it a good candidate for
distance
aBz/az
determination. FIG. 10B depicts a plot of the ratio 3 a2Briaz2 versus actual
distance. The
second spatial derivative in this ratio may also be determined by measuring
a/az (aBz/ar) or
aBr/ar
a/ar(aBz/az). FIG. 10C depicts a plot of the ratio 3 a2Briar2 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
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poles. The ratio in FIG. 10C shows very little sensitivity to the absolute
intensity of the
aBr/ar
magnetic poles. FIG. 10D depicts a plot of the ratio 1.5 a213,/3z 2 versus
actual distance. The
second spatial derivative in this ratio may also be determined by measuring
a/ar (aBz/az) or
a/az (aBz/ar). 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,
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substitutions and alternations can be made herein without departing from the
spirit and scope
of the disclosure as defined by the appended claims.
