U.S. patent application number 16/418342 was filed with the patent office on 2019-09-05 for correction of motion effect in nuclear magnetic resonance (nmr) logging.
The applicant listed for this patent is Schlumberger Technology Corporation. Invention is credited to Martin Hurlimann, Shin Utsuzawa, Haitao Zhang, Lukasz Zielinski.
Application Number | 20190271224 16/418342 |
Document ID | / |
Family ID | 52587223 |
Filed Date | 2019-09-05 |
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United States Patent
Application |
20190271224 |
Kind Code |
A1 |
Utsuzawa; Shin ; et
al. |
September 5, 2019 |
CORRECTION OF MOTION EFFECT IN NUCLEAR MAGNETIC RESONANCE (NMR)
LOGGING
Abstract
A method for making NMR measurements includes, in one
embodiment, using an NMR tool to acquire NMR measurements that are
effected by relative motion of the NMR tool and/or the specimen
under investigation. The NMR tool may include a plurality of
permanent magnets and a plurality of radio frequency (RF) coils.
The relative motion is estimated and used to modify an NMR
inversion kernel which is in turn used to transform the NMR
measurements into motion-corrected NMR measurements. Corresponding
systems, devices, and apparatuses are also disclosed herein.
Inventors: |
Utsuzawa; Shin; (Arlington,
MA) ; Hurlimann; Martin; (Newton, MA) ;
Zielinski; Lukasz; (Arlington, MA) ; Zhang;
Haitao; (Sugar Land, TX) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Schlumberger Technology Corporation |
Sugar Land |
TX |
US |
|
|
Family ID: |
52587223 |
Appl. No.: |
16/418342 |
Filed: |
May 21, 2019 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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14914193 |
Feb 24, 2016 |
10301924 |
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PCT/US2014/052029 |
Aug 21, 2014 |
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16418342 |
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61869735 |
Aug 25, 2013 |
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Current U.S.
Class: |
1/1 |
Current CPC
Class: |
E21B 17/10 20130101;
E21B 47/024 20130101; G01V 3/32 20130101; G01V 3/14 20130101; G01V
3/38 20130101; E21B 47/00 20130101; G01V 3/18 20130101 |
International
Class: |
E21B 47/024 20060101
E21B047/024; G01V 3/38 20060101 G01V003/38; G01V 3/18 20060101
G01V003/18; E21B 47/00 20060101 E21B047/00; E21B 17/10 20060101
E21B017/10; G01V 3/32 20060101 G01V003/32; G01V 3/14 20060101
G01V003/14 |
Claims
1. A method for logging a subterranean wellbore, the method
comprising: (a) using a nuclear magnetic resonance (NMR) logging
tool deployed in a subterranean wellbore to acquire NMR logging
measurements, the NMR logging tool including a plurality of
permanent magnets and a plurality of radio frequency (RF) coils,
the NMR logging measurements being effected by relative motion of
the NMR logging tool in the wellbore; (b) estimating said relative
motion of the NMR logging tool in (a); (c) estimating a motion
effect from the relative motion estimated in (b); (d) modifying an
NMR inversion kernel with the motion effect estimated in (c) to
obtain a modified kernel; (e) inverting the NMR logging
measurements acquired in (a) using the modified kernel obtained in
(d) to compute motion corrected NMR logging data.
2. The method of claim 1, wherein the NMR logging tool comprises an
NMR logging while drilling tool.
3. The method of claim 1, wherein (b) comprises estimating said
relative motion of the NMR logging tool using at least one sensor
selected from accelerometers, magnetometers, gyroscopes, calipers,
or standoff sensors.
4. The method of claim 1, wherein (b) comprises modeling transient
dynamic behavior of the NMR logging tool in the wellbore.
5. The method of claim 1, wherein the motion effect comprises a
motion induced signal decay (MID).
6. The method of claim 5, wherein the MID comprises at least one of
a displacement-dependent signal attenuation component and a
velocity-dependent signal attenuation component.
7. The method of claim 5, wherein the MID comprises an exponential
decay.
8. The method of claim 5, wherein the MID comprises a sum of
multiple exponentials summed over a number of components in the
MID.
9. The method of claim 5, wherein the MID comprises a first
exponential decay having a corresponding first time constant and
the NMR inversion kernel comprises a second exponential decay
having a corresponding second time constant.
10. The method of claim 1, wherein modifying the NMR inversion
kernel in (d) comprises multiplying the NMR inversion kernel by a
correction factor obtained from the motion effect estimated in
(c).
11. The method of claim 1, wherein the inversion kernel comprises a
matrix of kernel elements with each row in the matrix corresponding
to an echo.
12. The method of claim 11, wherein: the motion-corrected NMR
logging data computed in (e) comprises a motion corrected T2
distribution; and the inversion kernel comprises an
NE.times.NT.sub.2 matrix, wherein NE represents a number of echoes
and NT.sub.2 represents a number of points in the T2
distribution.
13. The method of claim 1, wherein the motion-corrected NMR logging
data computed in (e) comprises at least one of a motion corrected
T1 distribution and a motion-corrected T2 distribution.
14. The method of claim 1, wherein the relative motion is a
relative lateral motion of the NMR logging tool in the
wellbore.
15. A method for making nuclear magnetic resonance (NMR)
measurements, the method comprising: (a) using a nuclear magnetic
resonance (NMR) tool to acquire NMR measurements of a specimen, the
NMR tool including a plurality of permanent magnets and a plurality
of radio frequency (RF) coils, the specimen undergoing relative
motion with respect to the NMR tool while the NMR measurements are
acquired; (b) estimating said relative motion in (a); (c)
estimating a motion induced signal decay (MID) from the relative
motion estimated in (b); (d) determining a motion effect kernel
(MEK) based on the MID estimated in (c); and (e) inverting the NMR
measurements acquired in (a) using the MEK determined in (d) to
compute motion corrected NMR data.
16. The method of claim 15, wherein the MID comprises at least one
of a displacement-dependent signal attenuation component and a
velocity-dependent signal attenuation component.
17. The method of claim 15, wherein the MEK is obtained by
multiplying an NMR inversion kernel by the MID.
18. The method of claim 17, wherein the MID comprises a first
exponential decay having a corresponding first time constant and
the NMR inversion kernel comprises a second exponential decay
having a corresponding second exponential decay.
19. The method of claim 15, wherein: the motion-corrected NMR
logging data computed in (e) comprises a motion corrected T2
distribution; and the MEK comprises an NE.times.NT2 matrix, wherein
NE represents a number of echoes and NT2 represents a number of
points in the T2 distribution.
20. The method of claim 15, wherein the motion-corrected NMR
logging data computed in (e) comprises at least one of a motion
corrected T1 distribution and a motion-corrected T2 distribution.
Description
CROSS REFERENCE TO RELATED APPLICATIONS
[0001] This application is a Continuation of U.S. patent
application Ser. No. 14/914,193, filed Feb. 24, 2016, which is a
national stage entry of Patent Cooperation Treaty Patent
Application PCT/US2014/052029, filed Aug. 21, 2014, which in turn
claims priority to U.S. Provisional Patent Application Ser. No.
61/869,735 filed Aug. 25, 2013, all with the same title.
BACKGROUND
Technical Field
[0002] The present disclosure relates generally to nuclear magnetic
resonance (NMR) logging and, more specifically, to techniques for
correction of motion effects in NMR logging.
Background Information
[0003] This section is intended to introduce the reader to various
aspects of art that may be related to various aspects of the
subject matter described and/or claimed below. This discussion is
believed to be helpful in providing the reader with background
information to facilitate a better understanding of the various
aspects of the present disclosure. Accordingly, it should be
understood that these statements are to be read in this light, not
as admissions of prior art.
[0004] Logging tools have long been used in wellbores to make, for
example, formation evaluation measurements to infer properties of
the formations surrounding the borehole and the fluids in the
formations. Common logging tools include electromagnetic tools,
nuclear tools, acoustic tools, and nuclear magnetic resonance (NMR)
tools, though various other types of tools for evaluating formation
properties are also available.
[0005] Early logging tools were run into a wellbore on a wireline
cable after the wellbore had been drilled. Modern versions of such
wireline tools are still used extensively. However, as the demand
for information while drilling a borehole continued to increase,
measurement-while-drilling (MWD) tools and logging-while-drilling
(LWD) tools have since been developed. MWD tools typically provide
drilling parameter information such as weight on the bit, torque,
temperature, pressure, direction, and inclination. LWD tools
typically provide formation evaluation measurements such as
resistivity, porosity, NMR distributions, and so forth. MWD and LWD
tools often have characteristics common to wireline tools (e.g.,
transmitting and receiving antennas, sensors, etc.), but MWD and
LWD tools are designed and constructed to endure and operate in the
harsh environment of drilling.
[0006] In LWD operations, the drilling process can induce a complex
lateral motion whose amplitude and frequency spectrum can depend on
a number of parameters. For instance, the motion can have random
and periodic components depending on various parameters, such as
weight-on-bit (WOB), RPM, stabilizer size, torque-on-bit (TOB),
and/or inclination, to name just a few example. Further, the motion
may also differ based on the drilling path orientation/direction,
i.e., vertical drilling and horizontal drilling may yield different
induced motion behavior.
[0007] NMR tools using in well logging typically measure, among
other things, relaxation times, such as transverse relaxation times
(T.sub.2), of formation fluids, which can range from a fraction of
a millisecond to several seconds. With respect to NMR logging
tools, typically, an excitation slice is determined by an
excitation bandwidth and a received slice is determined by a
receiver bandwidth. A sensitive region may be determined based upon
the smaller of excitation bandwidth (usually depends on available
RF power) and receiver bandwidth. Essentially, the sensitive region
is the overlap between the excited slice and the received slice,
usually having the shape of a concentric shell. If an NMR logging
tool moves by a sizeable fraction of the excited slice (typically
having a thickness on the order of 1 centimeter) during tool
operation, the resulting measurements can have reduced accuracy. As
an example, the influence of tool motion can appear as an
additional signal decay that makes an apparent T.sub.2 appear
shorter than its intrinsic value (e.g., expected value if no
induced motion were present). This can result in an
under-estimation of permeability, which is used to evaluate
formation productivity. Accordingly, addressing the effects of tool
measurements that can be caused by the above-described types of
induced lateral motion during LWD drilling applications is a
challenge for the industry. It would be desirable to have a
technique for removing or otherwise compensating for such motion
effects.
SUMMARY
[0008] A summary of certain embodiments disclosed herein is set
forth below. It should be understood that these aspects are
presented merely to provide the reader with a brief summary of
certain embodiments and that these aspects are not intended to
limit the scope of this disclosure. Indeed, this disclosure may
encompass a variety of aspects that may not be set forth in this
section.
[0009] In accordance with one example embodiment, a method for
logging a subterranean wellbore is disclosed. The method includes
using a nuclear magnetic resonance (NMR) logging tool deployed in a
subterranean wellbore to acquire NMR logging measurements. The NMR
logging tool includes a plurality of permanent magnets and a
plurality of radio frequency (RF) coils. The NMR logging
measurements are effected by relative motion of the NMR logging
tool in the wellbore. The relative motion of the NMR logging tool
is estimated. The corresponding motion effect is also estimated. An
NMR inversion kernel is modified based on the motion effect to
obtain a modified kernel. The NMR logging measurements are then
inverted using the modified kernel to compute motion corrected NMR
logging data such as a T1 and/or a T2 distribution.
[0010] In accordance with another example embodiment, a method for
making nuclear magnetic resonance (NMR) measurements is disclosed.
The method includes using an NMR tool to acquire NMR measurements
of a specimen that undergoes relative motion with respect to the
NMR tool while the measurements are acquired. The NMR tool includes
a plurality of permanent magnets and a plurality of radio frequency
(RF) coils. The relative motion is estimated and used to further
estimate a motion induced signal decay (MID). A motion effect
kernel (MEK) is determined based on the MID and is in turn used to
invert the NMR measurements to compute motion-corrected NMR
data.
[0011] Again, the brief summary presented above is intended to
familiarize the reader with certain aspects and contexts of
embodiments of the present disclosure without limitation to the
claimed subject matter.
BRIEF DESCRIPTION OF DRAWINGS
[0012] The present disclosure is best understood from the following
detailed description when read with the accompanying figures. It is
emphasized that, in accordance with standard practice in the
industry, various features are not necessarily drawn to scale. In
fact, the dimensions of various features may be arbitrarily
increased or recued for clarify of discussion.
[0013] FIG. 1 is a schematic diagram of a wellsite system that may
be used for implementation of an example embodiment.
[0014] FIG. 2 is an example embodiment of a nuclear magnetic
resonance (NMR) logging tool that may be used in the wellsite
system of FIG. 1.
[0015] FIGS. 3A-3C are graphs that depict various types of lateral
tool motion that may be experienced by an NMR logging tool during
operation in a borehole.
[0016] FIG. 4 is a simplified example showing how lateral
displacement of an NMR tool within a borehole during operation can
affect NMR measurements.
[0017] FIG. 5 is an example embodiment of a method for obtaining
motion-corrected NMR data using a motion-effect kernel.
[0018] FIG. 6 shows an example embodiment of a technique for
estimating motion- induced signal decay using spin dynamics
simulation.
[0019] FIGS. 7A and 7B are graphs showing signal decay induced by
linear and circular motion, respectively, of a logging tool at
various amplitudes and frequencies.
[0020] FIG. 8 shows the effect of motion amplitude and frequency on
motion-induced decay, in accordance with an example embodiment.
[0021] FIG. 9 illustrates a motion-effect kernel that is used for
inversion of motion-affected NMR data, in accordance with an
example embodiment.
[0022] FIG. 10 shows an example of the result of inversion of
motion-affected NMR using the motion-effect kernel for a tool
undergoing a modest degree of motion.
[0023] FIG. 11 shows an example of the result of inversion of
motion-affected NMR using the motion-effect kernel for a tool
undergoing more severe motion when compared to FIG. 10
[0024] FIG. 12 is another example embodiment of a method for
obtaining motion-corrected NMR data using a motion-effect
kernel.
DETAILED DESCRIPTION
[0025] One or more specific embodiments of the present disclosure
are described below. These embodiments are merely examples of the
presently disclosed techniques. Additionally, in an effort to
provide a concise description of these embodiments, all features of
an actual implementation may not be described in the specification.
It should be appreciated that in the development of any such
implementation, as in any engineering or design project, numerous
implementation-specific decisions are made to achieve the
developers' specific goals, such as compliance with system-related
and business-related constraints, which may vary from one
implementation to another. Moreover, it should be appreciated that
such development efforts might be complex and time consuming, but
would nevertheless be a routine undertaking of design, fabrication,
and manufacture for those of ordinary skill having the benefit of
this disclosure.
[0026] When introducing elements of various embodiments of the
present disclosure, the articles "a," "an," and "the" are intended
to mean that there are one or more of the elements. The embodiments
discussed below are intended to be examples that are illustrative
in nature and should not be construed to mean that the specific
embodiments described herein are necessarily preferential in
nature. Additionally, it should be understood that references to
"one embodiment" or "an embodiment" within the present disclosure
are not to be interpreted as excluding the existence of additional
embodiments that also incorporate the recited features.
[0027] FIG. 1 represents a simplified view of a well site system in
which various embodiments can be employed. The well site system
depicted in FIG. 1 can be deployed in either onshore or offshore
applications. In this type of system, a borehole 11 is formed in
subsurface formations by rotary drilling in a manner that is well
known to those skilled in the art. Some embodiments can also use
directional drilling.
[0028] A drill string 12 is suspended within the borehole 11 and
has a bottom hole assembly (BHA) 100 which includes a drill bit 105
at its lower end. The surface system includes a platform and
derrick assembly 10 positioned over the borehole 11, with the
assembly 10 including a rotary table 16, kelly 17, hook 18 and
rotary swivel 19. In a drilling operation, the drill string 12 is
rotated by the rotary table 16 (energized by means not shown),
which engages the kelly 17 at the upper end of the drill string.
The drill string 12 is suspended from a hook 18, attached to a
traveling block (also not shown), through the kelly 17 and a rotary
swivel 19 which permits rotation of the drill string 12 relative to
the hook 18. As is well known, a top drive system could be used in
other embodiments.
[0029] Drilling fluid or mud 26 may be stored in a pit 27 formed at
the well site. A pump 29 delivers the drilling fluid 26 to the
interior of the drill string 12 via a port in the swivel 19, which
causes the drilling fluid 26 to flow downwardly through the drill
string 12, as indicated by the directional arrow 8 in FIG. 1. The
drilling fluid exits the drill string 12 via ports in the drill bit
105, and then circulates upwardly through the annulus region
between the outside of the drill string 12 and the wall of the
borehole, as indicated by the directional arrows 9. In this known
manner, the drilling fluid lubricates the drill bit 105 and carries
formation cuttings up to the surface as it is returned to the pit
27 for recirculation.
[0030] The drill string 12 includes a BHA 100. In the illustrated
embodiment, the BHA 100 is shown as having one MWD module 130 and
multiple LWD modules 120 (with reference number 120A depicting a
second LWD module 120). As used herein, the term "module" as
applied to MWD and LWD devices is understood to mean either a
single tool or a suite of multiple tools contained in a single
modular device. Additionally, the BHA 100 includes a rotary
steerable system (RSS) and motor 150 and a drill bit 105.
[0031] The LWD modules 120 may be housed in a drill collar and can
include one or more types of logging tools. The LWD modules 120 may
include capabilities for measuring, processing, and storing
information, as well as for communicating with the surface
equipment. By way of example, the LWD module 120 may include a
nuclear magnetic resonance (NMR) logging tool, and may include
capabilities for measuring, processing, and storing information,
and for communicating with surface equipment.
[0032] The MWD module 130 is also housed in a drill collar, and can
contain one or more devices for measuring characteristics of the
drill string and drill bit. In the present embodiment, the MWD
module 130 can include one or more of the following types of
measuring devices: a weight-on-bit measuring device, a torque
measuring device, a vibration measuring device, a shock measuring
device, a stick/slip measuring device, a direction measuring
device, and an inclination measuring device (the latter two
sometimes being referred to collectively as a D&I package). The
MWD tool 130 further includes an apparatus (not shown) for
generating electrical power for the downhole system. For instance,
power generated by the MWD tool 130 may be used to power the MWD
tool 130 and the LWD tool(s) 120. In some embodiments, this
apparatus may include a mud turbine generator powered by the flow
of the drilling fluid 26. It is understood, however, that other
power and/or battery systems may be employed.
[0033] The operation of the assembly 10 of FIG. 1 may be controlled
using control system 152 located at the surface. The control system
152 may include one or more processor-based computing systems. In
the present context, a processor may include a microprocessor,
programmable logic devices (PLDs), field-gate programmable arrays
(FPGAs), application-specific integrated circuits (ASICs),
system-on-a-chip processors (SoCs), or any other suitable
integrated circuit capable of executing encoded instructions
stored, for example, on tangible computer-readable media (e.g.,
read-only memory, random access memory, a hard drive, optical disk,
flash memory, etc.). Such instructions may correspond to, for
instance, workflows and the like for carrying out a drilling
operation, algorithms and routines for processing data received at
the surface from the BHA 100 (e.g., as part of an inversion to
obtain one or more desired formation parameters), and so forth.
[0034] Before discussing the motion correction techniques set forth
in this disclosure, some background with respect to the operation
of NMR logging tools is first provided. NMR well logging tools are
typically used to measure the properties of nuclear spins in the
formation, such as the longitudinal (or spin-lattice) relaxation
time (usually referred to as T.sub.1), transverse (or spin-spin)
relaxation time (usually referred to as T.sub.2), and diffusion
coefficient (D). Knowledge of these NMR properties can help aid in
determination of basic formation properties such as permeability
and porosity, as well as the fluid properties such as fluid type
and viscosity.
[0035] By way of background, NMR logging tools, i.e., LWD tool 120
of FIG. 1, may use permanent magnets to create a strong static
magnetic polarizing field inside the formation. The hydrogen nuclei
of water and hydrocarbons are electrically charged spinning protons
that create a weak magnetic field, similar to tiny bar magnets.
When a strong external magnetic field from the logging tool passes
through a formation containing fluids, these spinning protons align
themselves like compass needles along the magnetic field. This
process, called polarization, increases exponentially with T.sub.1
(longitudinal relaxation time), while the external magnetic field
(usually referred to as the B.sub.0 field) is applied.
[0036] FIG. 2 shows an example of an NMR logging tool 40 that is
described in commonly assigned U.S. Pat. No. 6,566,874, which is
hereby incorporated by reference. As an example, the illustrated
device in FIG. 2 may be used as the LWD tool 120 or part of an LWD
tool suite 120A. The NMR tool 40 may be constructed to conduct two
different measurements in two different locations using two
different gradients. For instance, sensitivity to motion may be
varied by varying the size of the resonance region by measuring in
different field geometries (i.e., a saddle point geometry and
gradient geometry, as an example of two different geometries) or by
performing measurements with different gradients. In this manner,
the NMR tool 40 may include upper 44, middle 46 and lower 48
permanent magnets that circumscribe an inner protective sleeve 60
of the NMR tool 40. The upper 44 and middle 46 magnets produce a
radial, axisymmetric static B.sub.0 field, and the middle 46 and
lower 48 magnets produce another radial, axisymmetric static
B.sub.0 field. Because, as an example, the upper 44 and middle 46
magnets are closer together than the middle 46 and lower 48
magnets, the upper B.sub.0 field has a higher gradient (and thus,
is more sensitive to motion) than the lower B.sub.0 field.
[0037] Among the other features of the illustrated NMR tool 40 are
that the tool 40 may include a radio frequency (RF) coil 54 which
acts as an antenna to transmit B.sub.1 pulses and receive spin echo
signals for the upper Bo field and an RF coil 56 to transmit
B.sub.1 pulses and receive spin echo signals for the lower B.sub.0
field. The coils 54 and 56 may be coupled to electronic circuitry
42 (of the NMR tool 40) that includes, among other things, B.sub.1
pulse generators 43 and a memory 45 to store indications of the
received spin echoes before transmitting indications of the spin
echoes uphole. The electronic circuitry 42 may be coupled to a
motion device 41 (i.e., an accelerometer, strain gauge, ultrasonic
finder and/or a magnetometer, as just a few examples) that
indicates motion of the NMR tool. This indication may be further
processed by the electronic circuitry 42 before being transmitted
uphole in some embodiments.
[0038] In operation, measurements are obtained by applying a second
oscillating magnetic field (usually referred to as the B.sub.1
field) as a series of pulses from an antenna (e.g., coil 54 in FIG.
2) of the NMR tool, which can be followed by or interleaved with
data acquisition. These pulses may be based on the
Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence or its variants, in
which trains of spin echoes are generated by a series of pulses.
The pulses cause the aligned protons to tip into a plane
perpendicular (transverse) to the direction of the polarization
field (B.sub.0). These tipped protons will start to precess around
the direction of the strong logging-tool magnetic field at a
frequency called the Larmor frequency.
[0039] The precessing protons create an oscillating magnetic field,
which generates weak radio signals at this frequency. The total
signal amplitude from the precessing hydrogen nuclei (e.g., a few
microvolts) is a measure of the total hydrogen content, or
porosity, of the formation. The rate at which the precession decays
is the transverse relaxation time (T.sub.2), which is indicative of
the rate at which the spinning protons lose their alignment within
the transverse plane. It can depend on certain factors, such as:
the intrinsic bulk-relaxation rate in the fluid; the
surface-relaxation rate, which is an environmental effect; and
relaxation from diffusion in a polarized field gradient, which is a
combination of environmental and tool effects.
[0040] Additionally, diffusion coefficients (D) can also be
measured by applying a set of pulses with variable durations in
between to encode the diffusive attenuation in spin echo
amplitudes. Further, NMR measurement types can be combined to
obtain information regarding the formation and/or the fluids
present therein. For instance, T.sub.2 and D measurements can be
combined to obtain two-dimensional information on formation fluids.
In another example, T.sub.2 and T.sub.1 measurements can be
combined as well. In general, any NMR measurements including but
not limited to the above examples may be combined to obtain
multi-dimensional information on the formation or formation
fluids.
[0041] Once the desired NMR data is acquired, mathematical
inversion processes can be applied to produce the distribution of
measured properties that reflects the anisotropy of formation or
formation fluids. For example, the T.sub.2 distribution represents
the distribution of pore sizes within the formation, and the area
under T.sub.2 curve represents the porosity filled with formation
fluids. Interpretation of pore size distribution and logarithmic
mean T.sub.2 may be used for calculating various petrophysical
parameters, such as permeability and the amount of free/bound
fluid.
[0042] One commonly used inversion scheme for NMR well logging
tools is based on a non-negative least squares method to derive the
distribution of one or more measured properties. The method
includes the formulation of a forward model, commonly referred to
as a kernel or kernel function. For example, when using a CPMG
pulse sequence, the amplitude of the k-th echo (ignoring
polarization and diffusion effects) can be given as:
m k _ = j = 1 NT 2 a j e - k t e T 2 j ( 1 ) ##EQU00001##
where NT.sub.2 represents the number of components in the T.sub.2
distribution, a.sub.j and T.sub.2j represent the amplitude and
relaxation time, respectively, of component j, and t.sub.e
represents the echo spacing. The overbar over the magnetization
symbol m indicates it is reconstructed.
[0043] The kernel function K can be an NE.times.NT.sub.2 matrix,
whose elements in the simplest case can be expressed as:
K kj = e - k t e T 2 j ( 2 ) ##EQU00002##
[0044] In Equation (2), the kernel function K.sub.kj shows the
response of the k-th echo at a decay rate of T.sub.2j. Each row in
the matrix corresponds to an echo and includes information on how
it responds to decay rates. When the NMR data contains other
sources of signal decay in addition to those attributable to
T.sub.2 relaxation, such as T.sub.1 relaxation or diffusion (D),
the above-mentioned kernel may be extended to accommodate such
additional decays. These additional decays may be analytically
formulated based on tool design and pulse sequence used.
[0045] Referring above to Equation (1), when written in matrix
form, Equation (1) can be rewritten as:
m=Ka (3)
[0046] The inversion process typically minimizes an objective
function .epsilon..sup.2, such as:
2 = k = 1 NE ( m k _ - m k ) 2 = ( m - K a ) T ( m - K a ) ( 4 )
##EQU00003##
where NE is the number of echoes in the CPMG echo train and the
superscript T means transpose (e.g., interchanging the rows and
columns of the matrix). The resulting solution, a, which is a
matrix of T.sub.2 components (a T.sub.2 distribution), can be given
by:
a=(K.sub.TK).sup.-1K.sup.Tm (5)
[0047] As will be appreciated, in some inversions of this type, a
non-negativity constraint can be applied to the components of a. In
addition, regularization and compression can also be applied.
[0048] As discussed above, one of the challenges with formation
evaluation using LWD operations is due to complex lateral motion
that is induced during the drilling process. For example, such
motion effects may have amplitude and frequency spectrums that
depend on a number of parameters. For instance, the motion can have
random and periodic components depending on various parameters,
such as weight-on-bit (WOB), rotations per minute (RPM) (or per
other unit of time, e.g., seconds), stabilizer size, torque-on-bit
(TOB), and/or wellbore inclination, to name just a few example.
Further, the motion behavior may also differ depending on whether
the drilling is occurring in a vertical or horizontal section of a
borehole. As an example, FIGS. 2A to 2C show several examples of
lateral tool motion that may negatively affect NMR measurements.
FIG. 3A depicts random motion with a relatively small amplitude.
FIG. 3B depicts a smooth forward whirl motion with a medium
amplitude. FIG. 3C depicts a rougher backward whirl motion with an
even larger amplitude, which is a situation that may occur when a
large WOB is applied in drilling vertically.
[0049] As described above, NMR measurements are typically made by
applying two magnetic fields, namely a static field (B.sub.0) and
an oscillating field (B.sub.1) to a specimen to measure nuclear
spin properties, the distributions of these magnetic fields may be
determined by tool geometry. Accordingly, if there is a net
relative displacement (other than diffusion) between the tool and
the specimen in inhomogeneous magnetic fields, the nuclear spins in
the specimen experience time-varying magnetic fields. These
magnetic field variations can cause signal attenuation, referred to
herein as "motion-induced decay" (MID) which can be generally
classified into two categories: (1) displacement-dependent signal
loss and (2) velocity-dependent signal loss.
[0050] With respect to displacement dependent signal loss, this is
due a limited sample volume observed by an NMR logging tool.
Measurement times for NMR logging are usually determined by the
measured property and a desired resolution. For example, intrinsic
T.sub.2 may range from a fraction of a millisecond to several
seconds. To have sufficient resolution to detect a particular
relaxation time, the measurement will usually have the duration of
the order of that targeted time. Therefore, where the
above-described lateral tool motion occurs over a millisecond to
several seconds, the measurement of the corresponding T.sub.2
component can be affected. With respect to T.sub.2 distributions,
the resulting effect from the signal decay caused by such lateral
tool motion is that the long T.sub.2 components are "squeezed" into
the shorter T.sub.2 side while maintaining the area under curve
(i.e., the porosity) constant. This can result in an
underestimation of certain formation parameters, such as
permeability.
[0051] A simplified example of this occurrence is shown in FIG. 4,
where an LWD NMR tool 120 operates while in the borehole 11 formed
in a formation 162. The left side of FIG. 4 shows a region (a
concentric shell) with in the formation 162 that is excited by the
tool 120 is depicted at 160. If during a time period that is on the
order of a targeted time (e.g., a millisecond to several seconds),
the tool 120 moves laterally in the borehole 11 to the position
shown on the right side of FIG. 4, it can be seen that the
detection region 164 (received slice) of the tool 120 does not
overlap fully with the excited region 160. This can negatively
affect the accuracy of the T.sub.2 measurement. The sensitive
region 165 is the overlap between the excited region 160 and
receiver region 164. The effect of the tool motion may appear as an
additional signal decay that makes the apparent T.sub.2 shorter
than the intrinsic value. As used herein "apparent" or the like
refers to the actual measurements obtained, and "intrinsic" or the
like refers to the measurements expected if the lateral tool motion
were absent.
[0052] If we consider a constant B.sub.0 gradient, g, and a
constant B.sub.1 over the sample volume, the excitation slice
thickness may be expressed as:
.DELTA. r = 2 B 1 g ( 6 ) ##EQU00004##
where .DELTA.r corresponds to the shell thickness of the excited
region 160 for a tool where the magnetic fields are axisymmetric
and the resulting sample volume is quasi-cylindrical shell(s). In
some tools, the sample volumes may have different shapes, such as
slabs, or other more complex shapes. When the displacement is
larger than a fraction of .DELTA.r, signal decay will occur
according to the amount of overlap between the sensitive region 164
and excited region 160. As an example and with reference to FIG.
7A, where displacement is a relatively small fraction of .DELTA.r,
such as 0.1.DELTA.r, at a low velocity (e.g., v=0.1 v.sub.ph) a
corresponding signal decay may be relatively small, i.e.,
approximately 5% at a displacement of 0.1 .DELTA.r. As the fraction
of .DELTA.r increases, such as to 0.5 .DELTA.r, and assuming the
same velocity, the signal decay also increases, i.e., to
approximately 30%. Generally speaking, a smaller overlap between
these regions can result in greater the signal loss. Additionally,
in some embodiments, the bandwidth of the NMR tool receiver (e.g.,
a receiving antenna) can also affect the signal loss. For example,
if the receiver bandwidth is less than the excitation bandwidth
(e.g., if region 164 is narrower than region 160 in FIG. 4), signal
loss may be less pronounced for the given displacement.
[0053] Velocity-dependent signal loss differs from
displacement-dependent signal loss in that it is due to the phase
shift acquired by spins moving in a magnetic field gradient. This
phase shift may correspond to the rotation of the effective
rotation axis around z-axis. It is analogous to applying pulses
with particular phases, which would deviate from the optimal phases
for the particular pulse sequence used. For example, the CPMG pulse
sequence yields a series of spin echoes by inverting the phase of
the spins with successive refocusing pulses, so that they
repeatedly pass through the points of the maximum coherence. Thus,
behavior of observed NMR signals depends on the phase of the
excitation and refocusing pulses applied. For example, consider a
spin that is moving at a constant velocity v in a magnetic field
with a linear gradient g. .gamma. represents the gyromagnetic ratio
of a nucleus of interest (e.g., .sup.1H). If perfect RF pulses are
assumed, an extra phase shift (in radians), .PHI., acquired at the
time of the first echo t=t.sub.E can be written as:
.phi. = .gamma. g vt E 2 4 ( 7 ) ##EQU00005##
[0054] If one assumes that the magnetization after the initial
90.degree..sub.x pulse lies exactly on the y-axis, then a
180.degree..sub.y pulse may behave like a pulse around an axis that
is shifted from the y-axis by .phi./2. Magnetization parallel to
the effective rotation axis behaves as in the CPMG sequence and
preserve the amplitude from echo to echo, while the component
perpendicular to the effective rotation axis behaves as in the
unmodified Carr-Purcell sequence and lead to the odd-even echo
oscillations with the overall signal decay, especially when there
are any pulse imperfections.
[0055] The above discussion is generally applicable to any system
where there is net relative displacement of spins with respect to
the B.sub.0 field, for example, when spins are stationary and the
tool is moving, or vice versa. More importantly, the signal loss is
in fact caused by the variation of the offset frequency
.DELTA..omega..sub.0=.omega..sub.0-.omega..sub.rf,, where
.omega..sub.0(r)=.gamma.B.sub.0(r) is the Larmor frequency
determined by the local B.sub.0 field at point r and the
gyromagnetic ratio .gamma. of the nucleus (e.g.,
.gamma.=2.pi..times.42.6 MHz/T for proton: .sup.1H), and
.omega..sub.rf is the tool's operating frequency at which the
signal transmission/reception takes place. Therefore, it is
possible to observe the same phenomena by creating a situation,
with each spin experiencing the variation of .DELTA..omega..sub.0.
By way of example, the above-referenced U.S. Pat. No. 6,566,874
discloses a method to mimic the effect of relative motion without
physically moving a tool or a sample. To observe odd-even echo
oscillations, the '874 patent discloses changing the tool operating
frequency (.omega..sub.rf) during NMR measurements.
[0056] The amount of the signal loss depends on the two regimes
mentioned above, as well as various parameters, but the observed
signal for the given motion-induced decay (MID) can be described in
a general form:
observed decay=intrinsic decayMID (8)
[0057] For example, when MID is defined as the sum of multiple
exponentials, the amplitude of k-th echo may be defined as in
accordance with Equation (1):
m k _ = j = 1 NT 2 a j e - k t e T 2 j l = 1 NT 2 m a l e k t e T 2
ml ( 9 ) ##EQU00006##
where NT.sub.2m is the number of components in the MID, and a.sub.1
and T.sub.2m1 are the amplitude and relaxation time of the
component 1 in the MID, respectively. It is noted that the
realization of MID is not limited to being exponential, but can
take any form depending on the nature of the tool motion and the
tool properties.
[0058] Having described the types of tool motion that may be
encountered in LWD operations, embodiments of the present
disclosure provide techniques for correcting motion-affected NMR
data obtained in during well logging. While the technique is
particularly beneficial to LWD applications where lateral tool
motion is generally a more prevalent issue, such techniques could
also be applied in wireline or slickline applications, i.e.,
correcting for motion experienced by a logging sonde used to log a
borehole by slickline or wireline. In accordance with aspects of
the disclosure, techniques for correcting motion-affected NMR data
may include compensating for signal decay introduced by a net
relative displacement between a sample (e.g., an excited region in
a formation) and a measuring apparatus (an LWD NMR tool 120) by
applying a correction factor to an inversion kernel. The correction
factor, in some embodiments, may be derived based on motion-induced
decay estimated by NMR spin dynamics simulation or a net relative
displacement (as a unit of excitation slice thickness (.DELTA.r),
i.e. an "effective displacement"), both being based on the relative
motion that is measured, modeled, or otherwise
predicted/determined. The details of such techniques are described
in further detail below.
[0059] A method 170 for correcting motion-affected NMR data is
depicted in FIG. 5 in accordance with an example embodiment. The
method 170 includes estimating relative motion at 172 for an NMR
logging tool. From the estimated relative motion, a motion-effect
can be estimated at 174. Using the motion-effect, an inversion
kernel can be derived at 176. The motion-effect kernel can then be
used for inverting NMR measurements acquired using the NMR logging
tool at 178.
[0060] In accordance with embodiments of this disclosure, the
estimation of relative motion (172) can be determined using sensors
located on the tool, by modeling the transient dynamic behavior of
the BHA (e.g., BHA 100 of FIG. 1), or using a combination of such
techniques. For instance, sensors that may be employed for
measuring tool motion may include one or a combination of an
accelerometer, magnetometer, gyroscope, caliper, or standoff
measurements. In some embodiments, such motion sensing/measurement
devices may be located proximate to the NMR sensor (e.g., the
magnet and/or antenna of an NMR tool), although in other
embodiments, such motion sensing/measurement devices may be located
further away from the NMR sensor (e.g., on a separate tool of the
BHA, such as in an MWD tool 130). It will be understood that
generally as the motion sensing device is placed further from the
NMR sensor, the resulting data may not accurately reflect tool
motion at the location for NMR sensor, which may cause the
resulting motion-induced decay to be over-estimated or
under-estimated.
[0061] With respect to BHA modeling, finite element analysis of the
BHA may be used. Modeling techniques may be employed to predict the
transient dynamic behavior of a BHA, such as by calculating the
interactions between the BHA and rock surfaces (i.e., accelerations
of the drill string collars when they impact the wellbore wall) for
given drilling parameters and formation properties. The output of
such modeling is the tool position at a given time step, from which
velocity, acceleration, and overall displacement may be derived. As
can be appreciated, BHA modeling can be used to complement downhole
motion measurements, or as an alternative if no downhole motion
measurement devices are available or such devices are located far
from the NMR sensor. In some embodiments, BHA modeling may also be
done using analytical solutions (e.g., without finite element
analysis). It will be appreciated that motion correction based on
measured, modeled, and/or predicted motion, in accordance with the
embodiments described herein, are applicable not only to LWD NMR
applications, but for any NMR applications where there is a net
relative displacement between a specimen and a measurement device.
Further, other types of modeling techniques may be also be used to
determine net relative displacement in various NMR applications,
and may be depend on the physics that governs such relative motion.
As examples, fluid dynamics modeling, particle dynamics modeling,
and/or net transport modeling may be used in some NMR applications
that encounter net relative displacement. Further, these types of
modeling techniques may, in some embodiments, be used instead or in
conjunction with BHA modeling for LWD NMR applications.
[0062] Once motion data is obtained (from 172), either as a
predicted specific trajectory or as parameters representing the
motion (e.g., overall amplitude and velocity) MID can be derived.
In one embodiment, MID can be derived using NMR spin-dynamics
solution, which can calculate the evolution of spins in given
magnetic fields at given time steps. Thus, by moving the magnetic
fields with respect to the nuclear spins, a spin dynamics
simulation can reproduce the effect of relative motion on NMR
measurements, where the amount of field shift at a given time is
obtained from the output of the previous step of estimating the
relative motion of the tool (e.g., 172 in FIG. 4). As can be
appreciated, this behavior is typically unique to each
tool/formation, and thus would typically be determined per tool,
per run.
[0063] FIG. 6 depicts generally a spin dynamics simulation
technique in accordance with an embodiment, in which spin dynamics
are calculated at each voxel of a Bloch vector map for spins at
each time step as a function of a time-varying B.sub.0 field and/or
B.sub.1 field (e.g., B0 maps, B1 maps, or both B0 and B1 maps may
be used). The B.sub.0 and/or B.sub.1 maps are updated at each time
step based on the estimated tool trajectory (e.g., from 172), and
the corresponding voxels in the B0 and/or B1 maps are used to
calculate spin rotation at each voxel of the spin maps. The maps
may have different values at each grid, which corresponds to the
inhomogeneity of the fields (B.sub.0 and/or B.sub.1).
[0064] In another embodiment, MID can be derived from net relative
displacement by considering special cases. For instance, in
accordance with Equation (7), to avoid appreciable phase shifts in
NMR signals:
.gamma. g vt E 2 4 << .pi. 2 ( 10 ) ##EQU00007##
which can be written as:
v ph << 2 .pi. .gamma. gt E 2 ( 11 ) ##EQU00008##
[0065] When above condition is satisfied, i.e., when motion is slow
compared to the echo spacing t.sub.E (i.e., fast pulsing regime),
then the on-resonance spins adiabatically track the effective
rotation axis to gradually get off-resonance. This process is
determined by the amplitude of motion and irrespective of the speed
of motion. Here, V.sub.ph represents the speed at which signal loss
can occur.
[0066] The relationship between displacement and signal loss can be
generalized by using the effective displacement in the unit of the
slice thickness .DELTA.r=2B.sub.1/g. FIGS. 7A and 7B show example
signal decays for linear (FIG. 7A) and circular (FIG. 7B)
trajectories as a function of the effective displacement with
various amplitudes and frequencies in a 1 G/cm magnetic field
gradient. The horizontal axes in FIGS. 7A and 7B represents the
effective displacement in the unit of the slice thickness
.DELTA.r=2B.sub.1/g. As can be seen, there are clear influences of
both the motion velocity and displacement on observed signal decay.
However, when velocity is much smaller than V.sub.ph, the decay
curves fall generally onto the same line regardless of the
realization of that velocity (i.e., combination of amplitude and
frequency). Further when velocity is sufficiently small (<0.1
v.sub.ph, curve 190 in FIG. 7A and curve 200 in FIG. 7B), linear
and circular motions possess nearly identical decay rate as a
function of the effective displacement up to approximately
.about..DELTA.r (the slice thickness of an excited
region/shell).
[0067] This allows for estimation of the amount of signal decay
based on the effective displacement (or other parameters correlated
with the effective displacement) without being dependent on knowing
the exact trajectories of relative motion. Therefore, although the
BHA modeling provides motion data that is not necessarily timely
correlated with NMR measurement, it is still viable as a source of
MID estimation.
[0068] It is noted that the observed signal decays in FIGS. 7A and
7B look similar to a Gaussian function, i.e., slow signal decay at
the beginning followed by more rapid decays. This is because (1)
the profile of the excitation slice is not a box function but
rather gradually decaying towards both ends of the slice and (2)
LWD NMR tools typically have axisymmetric magnetic fields and
resulting pseudo-cylindrical sensitive volume, which may result in
the size of the overlap region before and after the displacement is
not linear to the amount of the displacement.
[0069] Another property that characterizes MID is the signal
recovery. In the fast pulsing regime mentioned above (e.g., when
Equation (11) is satisfied), on-resonance spins adiabatically track
the effective rotation axis to gradually get off-resonance. If the
net displacement is sufficiently small, then a signal recovery can
be seen when motion brings those spins back into resonance. This is
shown in curve 220 of FIG. 8, which illustrates the effect of
motion amplitude on motion-induced decay. Here, each curve 220,
222, 224 represents a motion with the same nominal velocity, but
with different amplitude and frequency. The initial portions of the
curves (e.g., between approximately 0-75 ms) are similar, as decay
is determined by displacement. For a motion with small amplitude
(curve 220), the signal recovers when the tool comes back to the
original position after one period (e.g., approximately t=125 ms
for 8 Hz motion-curve 220). However, once displacement becomes
appreciable compared to the slice thickness, signal loss may be
irreversible, as shown by curve 224.
[0070] Next, once MID is determined (from 174), a correction can be
applied to an inversion kernel, which can be used to "remove"
motion effects from motion-affected NMR data. In accordance with
embodiments of this disclosure, a correction factor <MID> may
be applied to the inversion kernel of Equation (2). The kernel,
with the correction factor applied, may be rewritten as
follows:
K kj ' = e - k t e T 2 j e - k t e T 2 m ( 12 ) ##EQU00009##
where T.sub.2m is the time constant of the motion-induced decay
that it is represented by exponential decay. Essentially, the
modified motion-effect kernel (MEK) is an NMR inversion kernel that
takes motion effects into account when inverting NMR measurements.
In general, the MID can take any suitable form. Equation (12) may
be rewritten in a general form:
K kj ' = e - k t e T 2 j f m ( k t e ) ( 13 ) ##EQU00010##
With this MEK, the inversion process (e.g., 178 of FIG. 5) fits the
observed NMR signal to <(reconstructed) intrinsic decay>
times the estimated <MID>.
[0071] Essentially, <MEK> is equivalent to <MID> times
the <conventional kernel> (e.g., exp ((-kt.sub.E)/T.sub.2j)),
where <MID> can take any suitable form (e.g.,
<MID>=f(k, t.sub.E) for the k-th echo. Because both the
fitting function and fitted data include the MID term, the
resulting T.sub.2 distribution reflects the intrinsic signal decay
that is independent of MID. Although T.sub.2 measurements in well
logging have been described herein as an example of a measurement
to which the above-described motion-effect correction techniques
may be applied, those of ordinary skill in the art will recognize
the disclosed methods are suitable for use in any system for which
the motion can be modeled, measured, or otherwise predicted.
[0072] FIG. 9 shows an example of the motion-effect kernel (MEK)
prepared for the inversion of NMR data under a particular motion
trajectory. The solid lines (230-242) represent the original
sensitivity for given echo numbers, while dashed lines (230a-242a)
represent the corresponding echo numbers attenuated by the MID. The
MID reduces the sensitivity of later echoes when compared to the
original kernel K (Equation (2)). In the present example, the MID
is characterized by a single exponential with the time constant
T.sub.2m=250 ms. As can be seen with reference to curve 240, the
1024th echo (corresponding to 1024 ms) loses almost all the
sensitivity (comparing curve 240 to its corresponding curve 240a).
Conversely, curve 230, representing the 1st echo, loses almost no
sensitivity.
[0073] FIG. 10 shows the result of an example inversion for T.sub.2
using the MEK obtained as described above for a tool that undergoes
modest motion. Curve 250 represents the intrinsic T.sub.2
distribution, whereas curve 252 represents the apparent T.sub.2
distribution. As can be seen, the MID introduced additional signal
decay results in a T.sub.2 distribution that is distorted at the
longer T.sub.2 end (e.g., the values are heavily "squeezed" between
approximately 100 to 1000 ms). Curve 254 represents the corrected
T.sub.2 distribution obtained using the MEK which, as can be seen
in FIG. 10, is much closer to the intrinsic T.sub.2 distribution
250.
[0074] FIG. 11 shows the result of an example inversion for T.sub.2
using the MEK obtained as described above for a tool that undergoes
more severe motion. Curve 260 represents the intrinsic T.sub.2
distribution, whereas curve 262 represents the apparent T.sub.2
distribution. Curve 264 represents the corrected T.sub.2
distribution obtained using the MEK. As the MID goes to zero within
the data acquisition window, the MEK loses the sensitivity for the
higher echo numbers, which can distort the reconstruction. The
resulting distribution curve is imperfect, but still useful for
estimation of formation properties.
[0075] FIG. 12 depicts a method 270 that is a more detailed
embodiment of the method 170 of FIG. 5. As shown, the method 270
includes obtaining tool motion data (e.g., amplitude and trajectory
and/or tool trajectory) using, for example, BHA modeling and/or
motion measurement data (e.g., accelerometer, gyroscope, caliper,
etc.) at 272. Using the tool motion data obtained at 272, a
motion-induced decay (MID) is estimated at 274 using spin dynamics
simulation and/or net relative displacement. At 276, the determined
MID is used to derive a motion-effect kernel (MEK) as shown at 278.
Using the MEK 278, motion-affected NMR measurements 280 undergo an
inversion at 282. The result of the inversion is motion-corrected
NMR data (e.g., a distribution, such as T.sub.2 distribution) at
284.
[0076] As will be understood, the various techniques described
above and relating to the processing of NMR measurements for motion
correction due to lateral tool motion are provided as example
embodiments. Accordingly, it should be understood that the present
disclosure should not be construed as being limited to only the
examples provided above. Further, it should be appreciated that the
NMR processing techniques disclosed herein may be implemented in
any suitable manner, including hardware (suitably configured
circuitry), software (e.g., via a computer program including
executable code stored on one or more tangible computer readable
medium), or via using a combination of both hardware and software
elements. Further, it is understood that the various NMR
motion-correction techniques described may be implemented on a
downhole processor (e.g., a processor that is part of an NMR
logging tool), such that the inversion using the MEK to obtain
motion-corrected NMR data is performed downhole, with the results
sent to the surface by any suitable telemetry technique.
Additionally, in other embodiments, NMR measurements may be
transmitted uphole via telemetry, and the inversion of such
measurements may be performed uphole on a surface computer (e.g.,
part of control system 152 in FIG. 1).
[0077] Further, those of ordinary skill in the art will recognize
that the motion effects mentioned above is not limited to well
logging, but also arises in any NMR measurement or its variation
associated with imaging (magnetic resonance imaging: MRI) where
there is net displacement between a specimen for investigation and
the magnetic fields applied by an NMR device to the specimen. Other
applications of such techniques may include medical applications,
as well as industrial setups where the given sample is transported
by some sort of automated transport systems, such as belt conveyor.
Another example is the measurement of flowing samples such as the
liquid/granular flow in a pipe, vessel, and/or channel. Another
example is the scanning of industrial, architectural, agricultural,
or other products and/or natural resources by moving a tool
relative to such specimens in a manner not dissimilar to well
logging of a formation surrounding a borehole.
[0078] While the specific embodiments described above have been
shown by way of example, it will be appreciated that many
modifications and other embodiments will come to the mind of one
skilled in the art having the benefit of the teachings presented in
the foregoing description and the associated drawings. Accordingly,
it is understood that various modifications and embodiments are
intended to be included within the scope of the appended
claims.
* * * * *