Note : Les descriptions sont présentées dans la langue officielle dans laquelle elles ont été soumises.
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SYSTEM AND METHOD FOR POSITION AND ORIENTATION DETECTION OF A
DOWNHOLE DEVICE
Technical Field of the Invention
The present invention relates to a system and a method for identifying or
monitoring the orientation and position of a device, such as a tool, intended
to be
moved through a medium or to be stationary arranged in said medium, such as,
but
not limited to rock, soil or clay.
Background for the Invention
Several ways of identifying and monitoring the orientation and position of a
downhole body are known in the prior art. One example is to use a Universal
Bottom
Hole Orientation (UBHO) sub, sometimes referred to as a mule shoe sub, said
sub
being connected to a mechanical or electronic survey unit. The sub and the
survey
unit are then usually pumped down the drill rods. When the sub is landing it
should
lock onto the body that need to be orientated, in known position, establishing
a
reference point for the survey unit and through that make it possible to log
the
"body's orientation in the well bore. Sometimes the sub can be downhole while
a
matching unit such as a pin is connected to the survey unit.
US 4,094,360 discloses using a mule shoe for locating a direction indicating
instrument in axial alignment with a direction instrument assembly, a
deflecting tool
or deflecting drill bit where the axially lowermost constituent element of the
direction
instrument assembly is a mule shoe. Such a retrievable orientation unit can be
used
for many applications and contrary to stationary orientation systems, it has
the
.. advantage that it can either be left stationary on the tool or retrieved
from the hole.
The retrievable option is essential for operating for example directional core
barrels,
wedges and other tools where the orientation unit may need to be taken to
surface.
Use of a mule shoe has, however, several inherent disadvantages, such as:
- the mule shoe relies on that the shoe or shaped unit slide/rotate into a
.. matching shed connected to the body that are going to be oriented. To
complete the
connection can be difficult and some axial force is usually necessary to
complete the
operation and fulfil the required intention. As a result the sub often needs
an
additional force, by means of a hydraulic pressure, to ensure that it locks
into
position.
- the mule shoe can only be connected to the body in one specific position. If
the shoe or the matching shape is worn or meet tip to tip, a jam may occur and
the
unit becomes stuck or may act falsely as connected to the body
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- if using the mule shoe for orientation of directional drilling tools with an
independently rotating internal drive shaft, a locking mechanism or coupling
between
the drive shaft and the outer body of the drill is needed in order to obtain a
reference
point. This locking coupling is sensitive to malfunctioning and will usually
be very
difficult to operate when the hole gets deep, due to high torque. Moreover, it
is
difficult to know if the locking device or coupling is in open or closed mode.
References should also be made to WO 2013/0028075 regarding an eccentric
bushing assembly for a wireline-operated directional core barrel drill.
Finally references should also be made to US 2009/0056938 regarding
estimating orientation of a liner conveyed in a wellbore and deploying the
liner based
on such orientation.
Ills therefore a need for an orientation system that does not need the current
mule shoe. Moreover it is a need for an orientation system that can sense the
position of a point on the outer body of a device with respect to gravity,
independent
of the position of the point on the peripheral surface of the outer body at
any time.
Moreover, it is also a need for a system that does not depend on a locking
mechanism or coupling between the drive shaft and the outer body of a device
being
a drill or other tools with an independently running internal drive shaft. It
is also a
need for a orientation system comprising a survey unit that can be removed and
brought to surface when orientating tools or bodies that do not require
continuous
orientation, such as stationary tools, like a wedges or a valves etc., and a
orientation
system where the survey unit can retracted to surface if malfunction occur or
servicing is needed. Moreover, its also need for a system that can be brought
to
surface for downloading data, i.e. does not necessary relay on downhole cables
or
other systems for sending signals between the downhole orientation system and
surface.
Summary of the Invention
The main principle of the present invention is to provide an alternative
orientation system that does not need the UBHO (muleshoe), for determining the
circumferential position of a reference point on an outer body of a device at
any time
or at certain periods in time, and to collect such information, so that the
position of a
device can be determined and/or calculated. Such determinations or
calculations
may be real-time activities or logged for later downloads, on surface or
downhole,
and if desired, used for adjustment of the position of the device. The
adjustment may
be automatic, semi-automatic or manual.
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According to this principle, a reference point member may be arranged on an
outer element for example an external body or part of a structure, the well
bore, a
tube, casing or the like, while detectors or devices sensing the position of
the
reference point member and gravity are positioned on the device whose position
are
.. being identified. The term "outer element" should be understood to also
comprise the
situation where the reference point member is arranged on an outer body of the
device, and the detectors are arranged internally, for example associated with
an
inner element arranged inside the device. In both cases, the detectors sense
outwards.
It should be appreciated, however, that the reference point member and the
detectors may be configured in such way that the detectors sense inwards, i.e.
that
the reference point member is arranged on a centrally positioned extension of
a tool
body, for example arranged on a shaft or the like, with the detectors arranged
on a
separate cylinder, body or ring surrounding the reference point member.
Nevertheless a gravity detector will sense downwards.
In the drawings and the detailed description of an embodiment below, only the
embodiment having the reference point member arranged on an outer body of the
device and detectors arranged on or associated with an inner element,
preferably
retractable from the outer body, are shown and described, without thereby
intending
to limit the invention and scope of protection to such embodiment. Also the
embodiments comprising a solution where the detectors are arranged on an outer
element such as an outer ring surrounding a centrally arranged reference point
member is intended to be encompassed by the scope of protection.
An object of the present invention is to provide a system that can provide the
.. same degree of flexibility as the mule shoe, i.e. either be stationary or
retrieved for
determining the position of a movable device or a part of a tool with respect
to or
relative to the earth gravity, said movable device moving through a bore or a
hole or
being stationary, in any case without the use of a mule shoe.
An object of the invention is also to obtain a system that can be used for
obtaining information of the orientation of a core sample.
Another object of the present invention is to provide an alternative way of
determining the position of a device, moveably or stationary arranged in a
well bore.
Yet another object of the present invention is to provide a solution where the
sensing devices and electronics applied for determining the position of a
device
forms a part that can be a fixed part, an integral part, and/or a retrievable
part of a
device being stationary or moved through a medium, the device for example
being a
drill string or a tool.
4
In particular, but not exclusively, an object of the present invention is to
provide a retrievable system and a method for identifying and/or monitoring
the
orientation of a tool face relative to gravity, in a well bore or a hole,
based on the
position of a reference point member on an outer element.
It should be appreciated that another object of the invention is to provide a
system that may be used in connection with orientation of multiple types of
downhole tools and systems, including but not limited to any type of drilling
system,
rod orientation, packers, wedges, valves, branches, monitoring and controlling
systems.
Another object of the present invention is to provide a system and a method
for improving the identification or monitoring of the position of a device
moving
through or stationary in a bore or hole, such as a drill string or a tool, by
determining
the position of a reference point member on the device relative to gravity.
It should be appreciated that the system does not depend on a locking
mechanism or coupling between the drive shaft and the body of the device if
used
on for example a drill with an independently running internal drive shaft.
Moreover
the sensing devices and electronics can be removed and brought to surface,
leaving only the relatively low cost reference point member downhole.
Yet another object of the invention is to provide an orientation system giving
good readings close to vertical, something that is known to be a problem for
existing systems based on moveable weights, balls and similar to determine the
low
side of the device.
It should also be appreciated that the orientation system according to the
invention may in certain cases be seated permanently.
According to the present invention a system for identifying or monitoring the
orientation and position of a device, such as a tool is provided. The system
is
intended to be moved through or be stationary arranged in a medium, such as
rock,
and comprises an orientation unit comprising an outer element and an inner
element. The inner element may be retrievable. The system further comprises:
- a fixed reference point member arranged on one element of the
orientation unit;
- at least one first detector for at any time sensing and thus identifying the
position of the fixed reference point member, either directly or indirectly:
at least one second detector for sensing earth gravity;
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- device(s) for connecting the at least one first detector and the at
least one
second detector with a processor for retrieving collected data from said at
least one first detector and said at least one second detector, using such
data
from said at least one first detector and said at least one second detector
for
5 calculating and/or determining the rotational orientation of the fixed
reference
point member relative to gravity. The data may also be used to calculate the
inclination of the device or orientation of a core sample.
The first and second detectors are preferably arranged on another element of
the
orientation unit than the reference point member.
According to one embodiment of the system, the fixed reference point
member may be mechanically or physically fixed on the outer element, while
said at
least one first and at least one second detector are arranged on the inner
element. In
a preferred embodiment, the outer element is the outer body of the device.
According to a second embodiment of the invention, the fixed reference point
member may be fixed on a body centrally arranged on the device whose position
should be identified, while said at least one first and at least one second
detector is
arranged on a ring or tube surrounding the centrally arranged fixed referenced
point
member arranged inside the device.
The reference point member may comprise at least one magnet, the first
detector may comprise at least one magnetic field detector, preferably
magnetometer or hall effect sensor and the second detector may comprise at
least
one gravitation detector, such as inclinometer and/or accelerometer,
preferably tri-
axes accelerometer.
By "magnet" it is in this application meant permanent magnets, electro
magnets or magnetic fields made in any other way. In this application,
whenever the
word "magnet" is written it should be interpreted as a permanent magnet,
electro
magnet or a magnetic field made in any other way.
Moreover, the reference point member may be placed in fixed seat(s) on the
outer body of the device, intended to create an artificial magnetic field that
can be
detected by the first detector.
Further, the first and second detectors may be arranged on the inner element,
providing data for a rotational position of the outer body of the device, and
thereby
the rotational displacement during operation of the device; location of a body
face;
and/or position of the outer body by measuring location of the reference point
member, preferably a permanent magnet, with respect to earth gravity.
The inner element may comprise a separate rotatable alignment shaft, and a
magnet may be fixed to the rotatable shaft, said magnet will always align with
the
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reference point member on the outer element, as said reference point member
comprises one or more magnets. Thus the alignment shaft will rotate due to the
magnetic forces, and independently of the inner element. In another embodiment
magnets are not fixed to the alignment shaft, but the alignment shaft itself
is
magnetic, and will rotate to align with the reference point magnets.
It should be appreciated that at least one magnet may be fixed to the end of
the rotatable alignment shaft, providing a magnetic field that is sensed by a
magnetic
field detector such as a magnetometer or hall effect sensor, allowing the
detector to
sense the shaft orientation. Based on the position of the shaft, the position
of the
reference point member will be given.
Moreover, the reference point member may be chosen from one of a
radioactive source, laser, permanent magnet, electromagnet and Radio Frequency
Identification (RF ID).
In one embodiment, the entire tool is made non-magnetic, allowing a
magnetometer also to read directions relative to magnetic north. The
magnetometer
may be the first detector, or another separate detector.
The present invention also relates to a method for identifying or monitoring
the
orientation and position of a device, such as a tool. The method relates to a
system,
intended to be moved through or left stationary in a medium, such as rock, and
comprises an orientation unit comprising an outer element and an inner
element.
The inner element is preferably retrievable. The method comprises the
following
steps:
- establishing a fixed reference point member mechanically or
physically on one
part of the orientation unit;
- arranging detectors or sensors on another part of the orientation unit;
- using at least one detector to sense signals emitted by the reference
point
member, thus at any time sensing and identifying the position of the fixed
reference point member;
- using at least one second detector for sensing earth gravity;
- connecting the output from at least one first detector and the at least one
second detector with a processor or retrieving collected data from said at
least
one first detector and said at least one second detector, using such data from
said at least one first detector and said at least one second detector for
calculating and/or determining the rotational orientation of the fixed
reference
point member relative to gravity. It may also be used to calculating and/or
determining the inclination of the body or orientation of a core sample.
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According to one embodiment of the invention, the method comprises fixing
the reference point member mechanically or physically to the outer element,
while
arranging said at least one first and at least one second detector on the
retrievable
inner element.
According to another embodiment, the method comprises fixing the reference
point member on a body centrally arranged on the device while arranging said
at
least one first and at least one second detector on a retrievable outer ring
surrounding the centrally arranged fixed reference point member.
According to one specific embodiment the method comprises:
- establishing a fixed reference point member on an outer element;
- arranging detectors on the inner element and/or associated elements
thereon;
- using the magnetic field from the reference point member, which is
mechanically or physically fixed on the outer element, as a reference for
detecting the position of the outer element;
- detecting at least the magnetic field by a first detector associated with
the
inner element in order to sense and thus identify the position of the
reference
point member on the outer element,
- using a second detector for detecting earth gravity; and
- communicating the detected information from the first detector and
the second
detector to a processor for calculating and/or determining the orientation of
the outer element.
The method may comprise arranging one or more magnets to a rotatable
alignment shaft, in the inner element of the orientation unit, allowing at
least one of
said magnet to align with a magnet on the outer element acting as reference
point
member, by rotating the alignment shaft, and arranging a magnet on an end of
said
alignment shaft, and using the magnetic field from it as a source for
monitoring and
identifying the position and orientation of the alignment shaft. In another
embodiment
magnets are not fixed to the alignment shaft, but the alignment shaft itself
is
magnetic, and will rotate to align with the reference point magnets. By using
the data
from the magnetic field detector and a gravitation detector, such as an
accelerometer
or inclinometer the position of the reference point member may be determined
relative to gravity.
Moreover, the method may also comprise determining or calculating real-time
activities or data can be logged for later downloads, on surface or downhole.
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Gravitation detectors, such as accelerometers or inclinometers are used
together with any one of the detectors for sensing the rotational position of
the outer
element, and thus the position for the device may be identified.
Moreover, it should also be appreciated that the system according to the
present invention is to provide an improved orientation system that can
replace the
current use of a mule shoe, and when operating a system with internal drive
shaft
that rotates independently of the tool body, it is not necessary nor required
to include
a locking device for aligning the mule shoe shed in a known position relative
to the
outer body.
According to the present invention, the detector for identifying the
rotational
position of the alignment shaft, when aligned with the reference point member,
may
in lieu of a magnet field detector, be provided at the circumference of the
free end of
the alignment shaft with a number of separated axially orientated sectors, co-
functioning with brushes or shoes, similar to the system used on the
commutator on
an electric motor, the position of the various sectors with respect to the
position of
the corresponding magnet on the alignment shaft being known.
Short Description of the Drawings
In the following principles and an embodiment of the invention shall be
described in further detail, referring to the drawings wherein:
Figure 1 discloses schematically one principle for a orientation unit
according
to the present invention, indicating a suitable position of an embodiment
forming the
reference point member and the configuration of the first and second detector;
Figure 2 discloses schematically and in principle for a orientation unit using
a
second embodiment of the invention, showing the device forming the reference
point
member, the detectors applied and a second one or more magnets fixed to a
rotatable alignment shaft;
Figure 3 shows schematically a directional core drill with the orientation
unit
according to the present invention for identifying or monitoring the
orientation and
position of a the drill;
Figure 4 shows mud motor drill unit, with the orientation unit according to
the
present invention for identifying or monitoring the orientation and position
of the unit
Figure 5 shows schematically a part of the interior of a drilling system, with
the
orientation unit according to the invention;
Figure 6 shows schematically a retrievable orientation unit in accordance to
the
invention.
9
Description of the Reference Signs
The following reference numbers and signs refer to the drawings:
Onentation unit
11 Outer element
12 Inner element
13 Magnet. reference point magnet
14 Centreline
Detector, Magnetic field detector
16 Detector. Accelerometer
17 Magnet, shaft alignment magnet
18 Magnet, shaft reference magnet
19 Rotatable alignment shaft
20, 20' Bearings
21 Alignment shaft extension
22 Drill bit
23 Deflection mechanism
24 Packer
-
Rod coupling
32 CPU. memory
34 Magnetic force
Detailed Description of the Embodiments shown in the Figures
5 The following description of the exemplary embodiments refers to the
accompanying drawings. The same reference numbers in different drawings
identify the same or similar elements. The following detailed description is
not
meant or intended to limit the invention. Moreover, the embodiments to be
discussed next are not limited to these configurations, but may be extended to
10 other arrangements as discussed later.
Reference throughout the specification to "one embodiment" or "an
embodiment" means that a particular feature, structure, or characteristic
described in connection with an embodiment is included in at least one
embodiment of the subject matter disclosed. Thus, the appearance of the
15 phrases In one embodiment" or "in an embodiment" in various places
throughout
the specification is not necessarily referring to the same embodiment.
Further,
the particular features. structures or
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characteristics may be combined in any suitable manner in one or more
embodiments.
Figure 1 discloses schematically a orientation unit showing the principle used
according to one embodiment of the present invention, indicating a suitable
position
5 of the magnet forming the reference point member 13 and the configuration
of the
first and second detector 15, 16. The Figure indicates schematically a
orientation unit
10 provided with an outer element 11 and a inner element 12 arranged inside
the
outer element 11. The orientation unit 10 constitutes as a separate but yet
integral
part of the drilling tool and is arranged downhole together with the tool.
According to
10 the element shown, a reference point member in form of a magnet 13 is
arranged in
a fixed position on the outer element 11, providing a magnetic field. The
outer
element 11 may have a cylindrical cross sectional shape, having a centreline
14. A
magnetic field detector such as a magnetometer or a halleffect sensor 15 may
be
incorporated in the inner element 12 and is preferably centrically positioned
with
respect to the inner element 12.The magnetic field detector 15 may be of any
known
type measuring the magnetic field of the magnet 13 arranged on the outer
element
11, and hence the circumferential position of the magnet 13. Moreover, the
orientation unit 10 also comprises a three-axis accelerometer 16 arranged on
the
inner element 12, the purpose of which is to sense the earth gravity.
The system according to the embodiment shown in Figure 1, functions in the
following way:
- The position of the magnet 13 fixed to the outer element 11 is known,
with
respect to, for example the eccentric mechanism or other features or
elements of the tool, since the position of the magnet 13 is physically and
mechanically fixed on the outer element 11, which in turn is fixed relative to
the tool.
- The position of the accelerometer 16 senses earth gravity and both
detectors
are connected to a processor 32, calculating the location of the magnet 13
relative to the earth gravity. The information from the detectors may either
be
processed and used in real time downhole or communicated to the surface, or
may be saved and processed when the inner element 12 is retrieved to the
surface. Knowing the magnet's 13 position relative to the earth gravity, it is
possible by means of processing to identify the position of e.g. the eccentric
mechanism relative to earth gravity, i.e. the tool face.
Figure 2 discloses schematically and in principle a further embodiment of the
invention, showing a second principle for a orientation unit 10, comprising
the
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magnet forming the reference point member 13, the two detectors 15,16 being
fixed
to an inner element 12. It should be appreciated that the magnet 13 is
mechanically
fixed to an outer element 11, and that the inner element 12 comprises a
rotatable
alignment shaft 19. The magnetic field detector 15 and the accelerometer 16
are
fixed to the inner element 12, the magnetic field detector 15 sensing the
orientation
position of the rotatable shaft 19, while the accelerometer 16 sensing the
earth
gravity. The orientation of the rotatable alignment shaft 19 has a relevance
to the
position of the reference point member 13.
The inner element 12 comprises a alignment shaft 19, rotatable suspended in
the inner element 12, allowing free rotation of the alignment shaft 19 with
respect to
the inner element 12. The bearings 20, 20' may for example be roller bearings.
According to the embodiment shown in Figure 2 the rotatable arranged shaft 19
is
provided with a alignment magnet 17. The alignment magnet 17 is arranged in
such
manner that the alignment magnet 17 always will align itself with the
reference point
magnet 13 fixed to the outer element 11 of the device due to the magnetic
field
forces 34 between the reference magnet 13 and alignment magnet 17.
According to the embodiment shown in Figure 2 the alignment shaft 19 is
provided with an extension 21, extending through the ball bearing 20'. At the
end
face of the end of the extension 21, a shaft reference magnet 18 is arranged.
Since
the shaft alignment magnet 17 always will align with the reference point
magnet 13
fixed to the outer element 11, the magnetic field of the shaft reference
magnet 18 will
rotate correspondingly, thus enabling the magnetic field detector 15 to sense
the
orientation of the alignment shaft 19, giving the orientation or placement of
reference
point magnet 13, which together with gravity, gives the orientation of the
device 10.
Compared to the embodiment shown in Figure 1, the embodiment shown in
Figure 2 will function in the following way:
- The reference point magnet 13 creates a magnetic field 34, and since
the
shaft alignment magnet 17 is fixed to the alignment shaft 19, the said
alignment shaft 19 will be rotated so that the magnets 13 and 17 will align at
all time. As a consequence of the alignment, the shaft reference magnet 18
located at the end of the alignment shaft extension 21will rotate
correspondingly, and any change in the magnetic field of the shaft reference
magnet 18 will be sensed by the magnetic field detector 15. In such way the
position of the reference point magnet 13 on the outer element 11 of the
device will be sensed.
- The accelerometer 16 will sense the orientation of the inner element
12
relative to the earth gravity and by means of a processor the orientation unit
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will be able to identify and monitor the orientation of the device, for
example the position of a deflection mechanism 23 relative to earth gravity.
- As for the embodiment disclosed in both Figure 1 and 2, the
information from
the detectors may either be processed and used real time downhole or
5 communicated to the surface, or may be saved and read after the inner
element 12 is retrieved to the surface.
Figure 3 shows schematically a directional core barrel drill tool with an
orientation unit 10 according to the present invention incorporated, for
identifying or
10 monitoring the orientation and position of the drilling tool. The tool
is provided with a
drill bit 22 connected to the internally arranged drive shaft (not shown). At
its
opposite end the tool is provided with a rod coupling 25 for connecting the
drive shaft
to a drill string (not shown), transferring rotation to the drive shaft and
the drill bit 22.
As shown the tool is also provided with a packer 24 for holding the
orientation unit 10
of the device fixed during drilling operation.
Figure 4 shows schematically a view of the exterior of mud motor drill
assembly and the orientation unit 10 according to the invention is shown in a
partly
section of the tool. As indicated, the tool comprises a deflection mechanism
23 for
changing the direction of drilling.
Figure 5 shows schematically in enlarged scale an orientation unit 10,
showing the retrievable inner element 12 according to the invention. As
indicated the
outer element 11 of the device is in the form of a tube and is provided with a
fixed
reference point member 13, which in the embodiment shown is a magnet, fixed to
the wall of the outer element 11. The detectors may be of a type and a
configuration
as disclosed above in Figures 1 and 2, i.e. magnetic field detector(s) 15 and
accelerometer 16, or of other types of detectors as discussed above in the
summary
of the invention.
In the figures, the box denoted A represents the electronics, processors and
sensing devices for sensing the position of the reference point member 13 on
the
outer element and gravity, for example in the form of a magnetic field
detector and
an accelerometer as disclosed either in Figure 1 or in Figure 2, or as further
defined
in the description above. The sensing devices may be arranged on a same
configuration as disclosed in Figure 1 or Figure 2.
Figure 6 shows schematically a retractable assembly for e.g. a mud motor,
.. directional core drill, wedge or similar device, The assembly comprises at
the least
an inner part of an orientation unit according to the present invention, the
inner part
will be retrieved together with the assembly. The inner part can either be
fastened to
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the assembly in a fixed position, or in a freely rotating position relative to
the
assembly. At one end the assembly is provided with a seating device 25, being
configured to make it possible to lock and release the assembly to the mud
motor,
directional core drill, wedge or similar device. Depending on the type of
device the
retractable assembly should be mounted to, the assembly may also comprise
other
detectors, sensors and equipment, samples or seating devices. This will be
obvious
to a person skilled in the art, based on knowledge to the device the assembly
is
mounted to.
If the fixed reference point member is in the form of one or more permanent
magnets, it may preferably, but not necessarily, be placed in fixed seats in
the tube
wall of the outer body of the device or element, creating an artificial
magnetic field
that can be sensed and read by a magnetic field detector.
It should be appreciated that the reference point member may either be
arranged as part of the surface of the outer element or device. Alternatively,
the
reference point member may be incorporated as an integral part of the wall of
outer
element. As a variant the entire or a portion of the outer element may be of a
magnetized material.
While the reference point members in one embodiment are magnets,
reference point member may be chosen from one of a radioactive source, laser,
permanent magnet, electromagnet and Radio Frequency Identification (RFID).
Systems such as electromagnets and RDIF are beneficial in that they can be
turned
off and thus one can avoid potential interference issues. The magnets,
typically
reference magnets, alignment magnets and shaft reference magnets can freely be
replaced with the above alternatives and still provide the same technical
effect. In a
sensor pair one can be an emitter such as a laser and the other be a sensed
part
such as a reflector. The parts can also be swapped in the above disclosure and
still
provide the same technical effect.
It should also be appreciated that the outer element and the inner elements do
not necessarily have to be coaxial, nor does the outer element have to fully
surround
the inner element, for instance as if it were a pipe. In fact reference units
such as
magnets can be inserted in the well outside the orientation unit, for instance
into the
cement during completion.
It should be appreciated that the algorithms applied for detecting a position
and an orientation of the downhole tool face and for creating an image of said
position and/or orientation, are based standard mathematics.
