Note: Descriptions are shown in the official language in which they were submitted.
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METHOD OF ANALYZING A SUBTERRANEAN FORMATION AND
METHOD OF PRODUCING A MINERAL HYDROCARBON FLUID
AND A COMPUTER READABLE MEDIUM
Field of the Invention
In one aspect, the present invention relates to a method of analyzing a
subterranean
formation traversed by a wellbore. In another aspect the invention relates to
a method of
producing a mineral hydrocarbon fluid from an earth formation. In still
another aspect, the
invention relates to a computer readable medium storing computer readable
instructions
that analyze one or more electromagnetic response signals.
Back2round of the Invention
In logging while drilling (LWD) geo-steering applications, it is advantageous
to
detect the presence of a formation anomaly ahead of or around a bit or bottom
hole
assembly. There are many instances where "Look-Ahead" capability is desired in
LWD
logging environments. Look-ahead logging comprises detecting an anomaly at a
distance
ahead of a drill bit. Some look-ahead examples include predicting an over-
pressured zone
in advance, or detecting a fault in front of the drill bit in horizontal
wells, or profiling a
massive salt structure ahead of the drill bit.
In U.S. Patent No. 5,955,884 to Payton et al, a tool and method are disclosed
for
transient electromagnetic logging, wherein electric and electromagnetic
transmitters are
utilized to apply electromagnetic energy to a formation at selected
frequencies and
waveforms that maximize radial depth of penetration into the target formation.
In this
transient EM method, the current applied at a transmitter antenna is generally
terminated
and a temporal change of voltage induced in a receiver antenna is monitored
over time.
When logging measurements are used for well placement, detection or
identification of anomalies can be critical. Such anomalies may include for
example, a
fault, a bypassed reservoir, a salt dome, or an adjacent bed or oil-water
contact.
U.S. patent applications published under numbers 2005/0092487, 2005/0093546,
2006/0038571, describe methods for localizing such anomalies in a subterranean
earth
formation employing transient electromagnetic (EM) reading. The methods
particularly
enable finding the direction and distance to a resistive or conductive anomaly
in a
formation surrounding a borehole, or ahead of the borehole, in drilling
applications.
Of the referenced U.S. patent application publications, US 2006/0038571 shows
that transient electromagnetic responses can be analyzed to determine
conductivity values
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of a homogeneous earth formation (single layer), and of two or three or more
earth layers,
as well as distances from the tool to the interfaces between the earth layers.
In principle, the methodology as set forth in US 2006/0038571 would work for
any
number of layers. However, the larger the number of layers, and particularly
when the
layers are thin, the more complicated the analysis is. For instance, a thinly
laminated
sand/shale sequence would be difficult to analyze employing the methodology as
set forth
in US 2006/0038571.
Summary of the Invention
In accordance with the invention there is provided a method of analyzing a
subterranean formation traversed by a wellbore, using a tool comprising a
transmitter
antenna and a receiver antenna, the subterranean formation comprising one or
more
formation layers and the method comprising:
suspending the tool inside the wellbore;
inducing one or more electromagnetic fields in the formation;
detecting one or more time-dependent transient response signals;
analyzing the one or more time-dependent transient response signals taking
into account
electromagnetic anisotropy of at least one of the formation layers.
The electromagnetic properties of a formation layer comprising a number of
thin
layers may be approximated by one formation layer comprising an
electromagnetic
anisotropy. It is thereby avoided to have to take into account each thin layer
individually
when inverting the responses.
Amongst other advantages of taking into account electromagnetic anisotropy, is
that anisotropy information is useful in precisely locating mineral
hydrocarbon fluid
containing reservoirs, as such reservoirs are often associated with
electromagnetic
anisotropy of formation layers.
Said method of analyzing a subterranean formation may be used in a geo-
steering
application wherein a geo-steering cue may be derived from the one or more
time-
dependent transient response signals, taking into account electromagnetic
anisotropy, and
wherein a drilling operation may be continued in accordance with the derived
geo-steering
cue in order to accurately place a well.
In another aspect there is provided a method of producing a mineral
hydrocarbon
fluid from an earth formation, the method comprising steps of:
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suspending a drill string in the earth formation, the drill string comprising
at least a
drill bit and measurement sub comprising a transmitter antenna and a receiver
antenna;
drilling a well bore in the earth formation;
inducing an electromagnetic field in the earth formation employing the
transmitter
antenna;
detecting a transient electromagnetic response signal from the electromagnetic
field, employing the receiver antenna;
deriving a geosteering cue from the electromagnetic response;
continue drilling the well bore in accordance with the geosteering cue until a
reservoir containing the hydrocarbon fluid is reached;
producing the hydrocarbon fluid.
In still another aspect, the invention provides a computer readable medium
storing
computer readable instructions that analyze one or more detected time-
dependent transient
electromagnetic response signals that have been detected by a tool suspended
inside a
wellbore traversing a subterranean formation after inducing one or more
electromagnetic
fields in the formation, wherein the computer readable instructions take into
account
electromagnetic anisotropy of at least one formation layer in the subterranean
formation.
Brief Description of the Drawings
The present invention is described in more detail below by way of examples and
with reference to the attached drawing figures, wherein:
FIG. 1A is a block diagram showing a system implementing embodiments of the
invention;
FIG. 1B schematically illustrates an alternative system implementing
embodiments
of the invention;
FIG. 2 is a flow chart illustrating a method in accordance with an embodiment
of
the invention;
FIG. 3 is a graph illustrating directional angles between tool coordinates and
anomaly coordinates;
FIG. 4A is a graph showing a resistivity anomaly in a tool coordinate system;
FIG. 4B is a graph showing a resistivity anomaly in an anomaly coordinate
system;
FIG. 5 is a graph illustrating tool rotation within a borehole;
FIG. 6 schematically shows directional components involving electromagnetic
induction tools relative to an electromagnetic induction anomaly;
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FIG. 7 is a graph showing the voltage response from coaxial V"(t), coplanar
V'(t),
and the cross-component V,(t) measurements for L = 1 m, for B= 30 , and a
distance D
m from a salt layer;
FIG. 8 is a graph showing the voltage response from coaxial V"(t), coplanar
V'(t),
5 and the cross-component V,(t) measurements for L = 1 m, for B= 30 , and a
distance D
100 m from a salt layer;
FIG. 9 is a graph showing apparent dip (Bapp(t)) for an arrangement as in FIG.
7;
FIG. 10 is a graph showing apparent conductivity (6pp(t)) calculated from both
the
coaxial (V,,(t)) and the coplanar (V.,(t)) responses for the same conditions
as in FIG. 9;
10 FIG. 11 is a graph showing apparent dip Bapp(t) for the L = 1 m tool
assembly when
the salt face is D = 10 m away, for various angles between the tool axis and
the target;
FIG. 12 is a graph similar to FIG. 11 whereby the salt face is D = 50 m away
from
the tool;
FIG. 13 is a graph similar to FIG. 11 whereby the salt face is D = 100 m away
from
the tool;
FIG. 14 is a schematic illustration showing a coaxial tool with its tool axis
parallel
to a layer interface;
FIG. 15 is a graph showing transient voltage response as a function of t as
given by
the coaxial tool of FIG. 14 in a two-layer formation at different distances
from the bed;
FIG. 16 is a graph showing the voltage response data of FIG. 15 in terms of
the
apparent conductivity (6pp(t));
FIG. 17 is similar to FIG. 16 except that the resistivities of layers 1 and 2
have been
interchanged;
FIG. 18 shows a graph of the 6pp(t) for the case D = 1 m and L = 1 m, for
various
resistivity ratios while the target resistivity is fixed at R2 = 1 S2m;
FIG. 19 shows a comparison of apparent conductivity at large values of t,
6,pp(t-.~-), for coaxial responses where D = 1 m and L = 1 m as a function of
conductivity
62 of the target layer while the local conductivity 61 is fixed at 1 S/m;
FIG. 20 graphically shows the same data as FIG. 19 plotted as the ratio of
target
conductivity over local layer conductivity 61 versus ratio of the late time
apparent
conductivity 6pp(t~-) over local layer conductivity 61;
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FIG. 21 shows a graph containing apparent conductivity (6pp(t)) versus time
for
various combinations of D and L;
FIG. 22 graphically shows the relationship between ray-path RP and transition
time
tc;
FIG. 23 is a schematic illustration showing a coaxial tool approaching or just
beyond a bed boundary;
FIG. 24 is a graph showing transient voltage response as a function of t as
given by
the coaxial tool of FIG. 23 at different distances D from the bed;
FIG. 25 is a graph showing the voltage response data of FIG. 24 in terms of
the
apparent conductivity (6pp(t));
FIG. 26 is similar to FIG. 25 except that the resistivities of layers 1 and 2
have been
interchanged;
FIG. 27 presents a graph comparing 6pp(t) of FIG. 25 and FIG. 26 relating to D
I m;
FIG. 28 shows a graph of 6pp(t) on a linear scale for various
transmitter/receiver
spacings L in case D = 50 m;
FIG. 29 graphically shows distance to anomaly ahead of the tool verses
transition
time (tc) as determined from the data of FIG. 25;
FIG. 30 schematically shows a coplanar tool approaching or just beyond a bed
boundary;
FIG. 31 is a graph showing transient voltage response data in terms of the
apparent
conductivity (6pp(t)) as a function of t as provided by the coplanar tool of
FIG. 30 at
different distances D from the bed;
FIG. 32 shows a comparison of the late time apparent conductivity (6pp(t--
>00)) for
coplanar responses where D = 50 m and L = 1 m as a function of conductivity 61
of the
local layer while the target conductivity 62 is fixed at 1 S/m;
FIG. 33 graphically shows the same data as FIG. 32 plotted as the ratio of
target
conductivity 62 over local layer conductivity 61 versus ratio of the late time
apparent
conductivity 6pp(t-->oo) over local layer conductivity 61;
FIG. 34 graphically shows distance to anomaly ahead of the tool verses
transition
time (tc) as determined from the data of FIG. 31;
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FIG. 35 schematically shows a model of a coaxial tool in a conductive local
layer
(1 S2m), a very resistive layer (100 S2m), and a further conductive layer (1
S2m);
FIG. 36 is a graph showing apparent resistivity response versus time, Rapp(t),
for a
geometry as given in FIG. 35 for various thicknesses A of the very resistive
layer;
FIG. 37 schematically shows a model of a coaxial tool in a resistive local
layer (10
S2m), a conductive layer (1 S2m), and a further resistive layer (10 S2m);
FIG. 38 is a graph similar to FIG. 36, showing apparent resistivity response
Rapp(t)
versus time for a geometry as given in FIG. 37 for various thicknesses A of
the conductive
layer;
FIG. 39 schematically shows a model of a coaxial tool in a conductive local
layer
(1 S2m) in the vicinity of a highly resistive layer (100 S2m) with a
separating layer having
an intermediate resistance (10 S2m) of varying thickness in between;
FIG. 40 is a graph similar to FIG. 36, showing apparent resistivity response
versus
time, Rapp(t), for a geometry as given in FIG. 39 for various thicknesses A of
the
separating layer;
FIG. 41 shows calculated coaxial transient voltage responses for an L = 1 m
tool in
an anisotropic formation wherein 6H = 1 S/m (RH = 1 S2m) for various values of
02;
FIG. 42 shows apparent conductivity based on the responses of FIG. 41;
FIG. 43 shows apparent conductivity based on coaxial responses for an L = 1 m
tool in a formation wherein 6H = 0.1 S/m for various values of (32;
FIG. 44 shows apparent conductivity based on coaxial responses for an L = 1 m
tool in a formation wherein 6H = 0.01 S/m for various values of (32;
FIG. 45 shows a graph plotting late time asymptotic value of coaxial apparent
conductivity 6Zz(t-->-) from FIGs. 44 to 44, normalized by o-H , against a
variable
representing (32;
FIG. 46 shows apparent dip angle 6app(t) as a function of time based on
calculated
coaxial, coplanar and cross-component transient responses from a L = 1 m tool
in a
formation of RH = 10 S2m and RV/RH = 9;
FIG. 47 shows an electromagnetic induction tool in a formation layer
comprising a
package of alternating sets of sub-layers;
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FIG. 48 shows a graph of apparent resistivity in co-axial measurement and co-
planar measurement of the geometry as in FIG. 47;
FIG. 49 schematically shows directional components of an electromagnetic
induction tool relative to an anisotropic anomaly;
FIG. 50 shows a plot of the apparent conductivity ((Tapp(z; t)) in both z- and
t-
coordinates for various distances D;
FIG. 51 shows a plot of the apparent conductivity ((Tapp(z; t)) in both z- and
t-
coordinates;
FIG. 52 schematically shows a model of a structure involving a highly
resistive
layer (100 S2m) covered by a conductive local layer (1 S2m) which is covered
by a resistive
layer (10 S2m), whereby a coaxial tool is depicted in the resistive layer;
FIG. 53A shows apparent resistivity in both z and t coordinates whereby
inflection
points are joined using curve fitted lines;
FIG. 53B shows an image log derived from FIG. 53A;
FIG. 54A schematically shows a coaxial tool seen as approaching a highly
resistive
formation at a dip angle of approximately 30 degrees;
FIG. 54B shows apparent dip response in both t and z coordinates for z-
locations
corresponding to those depicted in FIG. 54A.
Detailed Description of the Invention
The present invention will now be described in relation to particular
embodiments,
which are intended in all respects to be illustrative rather than restrictive.
Alternative
embodiments will become apparent to those skilled in the art to which the
present
invention pertains without departing from its scope.
It will be understood that certain features and sub-combinations are of
utility and
may be employed without reference to other features and sub-combinations
specifically set
forth. This is contemplated and within the scope of the claims.
Embodiments of the invention relate to analysis of electromagnetic (EM)
induction
signals and to a system and method for determining distance and/or direction
to an
anomaly in a formation from a location within a wellbore. The analysis is
sensitive to
electromagnetic anomalies, in particular electromagnetic induction anomalies.
Both frequency domain excitation and time domain excitation have been used to
excite electromagnetic fields for use in anomaly detection. In frequency
domain excitation,
a device transmits a continuous wave of a fixed or mixed frequency and
measures
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responses at the same band of frequencies. In time domain excitation, a device
transmits a
square wave signal, triangular wave signal, pulsed signal or pseudo-random
binary
sequence as a source and measures the broadband earth response. Sudden changes
in
transmitter current cause transient signals to appear at a receiver caused by
induction
currents in the formation. The signals that appear at the receiver are called
transient
responses because the receiver signals start at a first value after a sudden
change in
transmitter current, and then they decay (or increase) with time to a new
constant level at a
second value. The technique disclosed herein implements the time domain
excitation
technique.
As set forth below, embodiments of the invention propose a general method to
determine a direction from a measurement sub to a resistive or conductive
anomaly using
transient EM responses. As will be explained in detail, the direction to the
anomaly is
specified by a dip angle and an azimuth angle. Embodiments of the invention
propose to
define an apparent dip (6app(t)) and an apparent azimuth (cpapp(t)) by
combinations of
multi-axial, e.g. bi-axial or tri-axial, transient measurements. The true
direction, in terms of
dip and azimuth angles ({0, cp}), may be determined from the analysis of the
apparent
direction ({6app(t), Tapp(t)}). For instance, the apparent direction
({6app(t), Tapp(t)})
approaches the true direction ({0, cp}) as a time (t) increases, if the
anomaly has a high
thickness as seen from the tool.
Time-dependent values for apparent conductivity may be obtained from coaxial
and
coplanar electromagnetic induction measurements, and can respectively be
denoted as
6coaxial(t) and 6coplanar(t). Both read the conductivity in the total present
formation
around the tool. The 6app(t) and cpapp(t) both initially read zero when an
apparent
conductivity 6coaxial(t) and 6coplanar(t) from coaxial and coplanar
measurements both
read the conductivity of the formation surrounding the tool nearby. The
apparent
conductivity will be further explained below and can also be used to determine
the location
of an anomaly in a wellbore.
Whenever in the present specification the term "conductivity" is employed, it
is
intended to cover also its inverse equivalent "resistivity", and vice versa.
The same holds
for the terms "apparent conductivity" and "apparent resistivity".
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FIGs. 1A and 1B illustrate systems that may be used to implement the
embodiments of the method of the invention. A surface computing unit 10 may be
connected with an electromagnetic measurement too12 disposed in a wellbore 4.
In FIG. 1A, the too12 is suspended on a cable 12. The cable 12 may be
constructed
of any known type of cable for transmitting electrical signals between the
too12 and the
surface computing unit 10.
In FIG. 113, the tool is comprised in a measurement sub 11 and suspended in
the
wellbore 4 by a drill string 15. The drill string 15 further supports a drill
bit 17, and may
support a steering system 19. The steering system may be of a known type,
including a
rotatable steering system or a sliding steering system. The wellbore 4
traverses the earth
formation 5 and it is an objective to precisely direct the drill bit 17 into a
hydrocarbon fluid
containing reservoir 6 to enable producing the hydrocarbon fluid via the
wellbore. Such a
reservoir 6 may manifest itself as an electromagnetic anomaly in the formation
5.
Referring again to both FIGs. 1A and 1B, one or more transmitters 16 and one
are
more receivers 18 may be provided for transmitting and receiving
electromagnetic signals
into and from the formation around the wellbore 4. A data acquisition unit 14
may be
provided to transmit data to and from the transmitters 16 and receivers 18 to
the surface
computing unit 10.
Each transmitter 16 and/or receiver 18 may comprise a coil, wound around a
support structure such as a mandrel. The support structure may comprise a non-
conductive
section to suppress generation of eddy currents. The non-conductive section
may comprise
one or more slots, optionally filled with a non-conductive material, or it may
be formed out
of a non-conductive material such as a composite plastic. Alternatively, the
support
structure is coated with a layer of a high-magnetic permeable material to form
a magnetic
shield between the antenna and the support structure.
Each transmitter 16 and each receiver 18 may be bi-axial or even tri-axial,
and
thereby contain components for sending and receiving signals along each of
three axes.
Accordingly, each transmitter module may contain at least one single or multi-
axis antenna
and may be a 3-orthogonal component transmitter. Each receiver may include at
least one
single or multi-axis electromagnetic receiving component and may be a 3-
orthogonal
component receiver.
A tool/borehole coordinate system is defined as having x, y, and z axes. The z-
axis
defines the direction from the transmitter T to the receiver R. It will be
assumed hereinafter
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that the axial direction of the wellbore 4 coincides with the z-axis, whereby
the x- and y-
axes correspond to two orthogonal directions in a plane normal to the
direction from the
transmitter T to the receiver R and to the wellbore 4.
The data acquisition unit 14 may include a controller for controlling the
operation
of the too12. The data acquisition unit 14 preferably collects data from each
transmitter 16
and receiver 18 and provides the data to the surface computing unit 10. The
data
acquisition unit 14 may comprise an amplifier and/or a digital to analogue
converter, to
amplify the responses and/or convert to a digital representation of the
responses before
transmitting to the surface computing unit 10 via cable 12 and/or an optional
telemetry unit
13.
The surface computing unit 10 may include computer components including a
processing unit 30, an operator interface 32, and a tool interface 34. The
surface computing
unit 10 may also include a memory 40 including relevant coordinate system
transformation
data and assumptions 42, an optional direction calculation module 44, an
optional apparent
direction calculation module 46, and an optional distance calculation module
48. The
optional direction and apparent direction calculation modules are described in
more detail
in US patent application publication 2005/0092487 and need not be further
described here,
other than specifying that these optional modules may take into account
formation
anisotropy.
The surface computing unit 10 may include computer components including a
processing unit 30, an operator interface 32, and a tool interface 34. The
surface computing
unit 10 may also include a memory 40 including relevant coordinate system
transformation
data and assumptions 42, a direction calculation module 44, an apparent
direction
calculation module 46, and a distance calculation module 48. The surface
computing unit
10 may further include a bus 50 that couples various system components
including the
system memory 40 to the processing unit 30. The computing system environment
10 is
only one example of a suitable computing environment and is not intended to
suggest any
limitation as to the scope of use or functionality of the invention.
Furthermore, although
the computing system 10 is described as a computing unit located on a surface,
it may
optionally be located below the surface, incorporated in the tool, positioned
at a remote
location, or positioned at any other convenient location.
The memory 40 preferably stores one or more of modules 48, 44 and 46, which
may be described as program modules containing computer-executable
instructions,
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executable by the surface computing unit 10. Each module may comprise or make
use of a
computer readable medium that stores computer readable instructions for
analyzing one or
more detected time-dependent transient electromagnetic response signals that
have been
detected by a tool suspended inside a wellbore traversing a subterranean
formation after
inducing one or more electromagnetic fields in the formation. The instructions
may
implement any part of the disclosure that follows herein below.
For example, the program module 44 may contain computer executable
instructions
to calculate a direction to an anomaly within a wellbore. The program module
48 may
contain computer executable instructions to calculate a distance to an anomaly
or a
thickness of the anomaly. The stored data 42 may include data pertaining to
the tool
coordinate system and the anomaly coordinate system and other data for use by
the
program modules 44, 46, and 48. Preferably, the computer readable instructions
take into
account electromagnetic anisotropy of at least one formation layer in the
subterranean
formation.
For further details on the computing system 10, including storage media and
input/output
devices, reference is made to US patent application publication 2005/0092487.
Accordingly, additional details concerning the internal construction of the
computer 10
need not be disclosed in connection with the present invention.
FIG. 2 is a flow chart illustrating the procedures involved in a method
embodying
the invention. The illustrate procedures may start at S. Generally, in
procedure "Transmit
Signals" (A), the transmitters 16 transmit electromagnetic signals. In
procedure "Receive
Responses" (B), the receivers 18 receive transient responses. In procedure
"Process
Responses" (C), the system processes the transient responses. The procedures
may then
end at E and/or start again at S.
Procedure C may comprise determining a distance and/or a direction to the
anomaly may be determined. Procedure C may comprise creating an image of
formation
features based on the transient electromagnetic responses. Electromagnetic
anisotropy of at
least one of the formation layers may be taken into account.
FIGs. 3-6 illustrate the technique for implementing procedure C for
determining
distance and/or direction to the anomaly. FIGs. 6 and 41 to 49 illustrate how
electromagnetic anisotropy may be taken into account, e.g. in determining
distance and/or
direction to the anomaly.
Tri-axial Transient EM Responses
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FIG. 3 illustrates directional angles between tool coordinates and anomaly
coordinates. A transmitter coil T is located at an origin that serves as the
origin for each
coordinate system. A receiver R is placed at a distance L from the
transmitter. An earth
coordinate system, includes a Z-axis in a vertical direction and an X-axis and
a Y-axis in the
East and the North directions, respectively. The deviated borehole is
specified in the earth
coordinates by a deviation angle Be and its azimuth angle Vb. A resistivity
anomaly A is
located at a distance D from the transmitter in the direction specified by a
dip angle (Ba)
and its azimuth (Va).
In order to practice embodiments of the method, FIG. 4A shows the definition
of a
tool/borehole coordinate system having x, y, and z axes. The z-axis defines
the direction
from the transmitter T to the receiver R. The tool coordinates in FIG. 4A are
specified by
rotating the earth coordinates (X, Y, Z) in FIG. 3 by the azimuth angle (Vb)
around the
Z-axis and then rotating by Bb around the y-axis to arrive at the tool
coordinates (x, y, z).
The direction of the anomaly is specified by the dip angle (a) and the azimuth
angle (0)
where:
(1) cos 0_(bz = a) = cos Ba cos Bb + sin Ba sin Bb cos(Va -Vb )
(2) tan 0 sin Bb sin(~pa - (pb )
cos Ba sin Bb cos(Va -Vb) - sin Ba cos Bb
Similarly, FIG. 4B shows the definition of an anomaly coordinate system having
a,
b, and c axes. The c-axis defines the direction from the transmitter T to the
center of the
anomaly A. The anomaly coordinates in FIG. 4B are specified by rotating the
earth
coordinates (X, Y, Z) in FIG. 3 by the azimuth angle (Va) around the Z-axis
and
subsequently rotating by Ba around the b-axis to arrive at the anomaly
coordinates (a, b, c).
In this coordinate system, the direction of the borehole is specified in a
reverse order by the
azimuth angle (0) and the dip angle (0).
Transient Responses in Two Coordinate Systems
The method is additionally based on the relationship between the transient
responses in two coordinate systems. The magnetic field transient responses at
the
receivers [Rx, Ry, Rz] which are oriented in the [x, y, z] axis direction of
the tool
coordinates, respectively, are noted as
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V~ Vxy Vxz Rx
(3) Vyx V,, Vyz = Ry [T T T],
V~ VZy Vu RZ
wherein the right-hand side of the equation represents all combinations of
receiver axis and
transmitter axis, whereby Vij = RiTj denotes voltage response sensed by
receiver Ri (i = x,
y, z) from signal transmitted by transmitter Tj (j = x, y, z). Each
transmitter may comprise
a magnetic dipole source, [Mx, My, Mz], in any direction.
When the resistivity anomaly is distant from the tool, the formation near the
tool is
seen as a homogeneous formation. For simplicity, the method may assume that
the
formation is isotropic. Only three non-zero transient responses exist in a
homogeneous
isotropic formation. These include the coaxial response and two coplanar
responses.
Coaxial response VZZ(t) is the response when both the transmitter and the
receiver are
oriented in the common tool axis direction. Coplanar responses, VXX(t) and
V55,(t), are the
responses when both the transmitter T and the receiver R are aligned parallel
to each other
but their orientation is perpendicular to the tool axis. All of the cross-
component responses
are identically zero in a homogeneous isotropic formation. Cross-component
responses are
either from a longitudinally oriented receiver with a transverse transmitter,
or vise versa.
Another cross-component response is also zero between a mutually orthogonal
transverse
receiver and transverse transmitter.
The effect of the resistivity anomaly is seen in the transient responses as
time
increases. In addition to the coaxial and the coplanar responses, the cross-
component
responses V,j(t) (i;;I; i, j = x, y, z) become non-zero.
The magnetic field transient responses may also be examined in the anomaly
coordinate system. The magnetic field transient responses at the receivers
[Ra, Rb, Rj that
are oriented in the [a, b, c] axis direction of the anomaly coordinates,
respectively, may be
noted as
V,, Vab Vac Ra
(4) Vba Vbb Vbc = Rb [Ta Tb Tc ]
Vca Vcb Vcc Rc
wherein the right-hand side of the equation represents all combinations of
receiver
orientation and transmitter orientation, whereby Vij = RiTi denotes voltage
response
sensed by receiver Ri (in orientation i = a, b, c) from signal transmitted by
transmitter Ti
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(in orientation j = a, b, c). Each transmitter may comprise a magnetic dipole
source, [Ma,
Mb, Mj, along the orientation a, b, or c.
When the anomaly is large and distant compared to the transmitter-receiver
spacing, the effect of spacing can be ignored and the transient responses can
be
approximated with those of the receivers near the transmitter. Then, the
method assumes
that axial symmetry exists with respect to the c-axis that is the direction
from the
transmitter to the center of the anomaly. In such an axially symmetric
configuration, the
cross-component responses in the anomaly coordinates are identically zero in
time-domain
measurements.
Vaa Vab Vac Vaa 0 0
(5) Vba Vbb Ubc - 0 Vaa 0
Vca Vcb Vcc 0 0 Vcc
The magnetic field transient responses in the tool coordinates are related to
those in the
anomaly coordinates by a simple coordinate transformation
P(0, 0) specified by the dip angle (0) and azimuth angle (0).
V. Vxy Vxz Vaa Vab Vac
(6) Vyx Vyy Vyz = P(T9,O)t" Vba Vbb Vbc P(0,0)
Vzx Vzy Vzz Vca Vcb Vcc
cos t9cos o cos t9sin o - sin O
(7) P(O, 0) _ - sin o cos O 0
sin 0cos o sin 0sin o cos 0
Determination of Direction
The assumptions set forth above contribute to determination of target
direction,
which is defined as the direction of the anomaly from the origin. The tool is
in the origin.
When axial symmetry in the anomaly coordinates is assumed, the transient
response
measurements in the tool coordinates are constrained and the two directional
angles may be
determined by combinations of tri-axial responses.
V. Vxy Vxz Vaa 0 0
(8) Vyx Vyy Vyz - P(O,0)t" 0 Vaa 0 P(O,0)
VZI Vzy VZZ 0 0 V"
In terms of each tri-axial response
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V~ = (Vaa COS 2 0 + V,, sm 2 g) COS 2 0 + Vaa sin 20
(9) Vyy =(Vaa cos 2 0 + V,, sin 2 0) sin z O+ Vaa cos z 0
V,, = Vaa sm 2 tg + V" COS 2 tg
VIY = Vyx = -(Vaa - V,,) sin 2 Ocos osin o
(10) V, =Vxz =-(Vaa -V,,)cosOsinOcos0
Vyz = Vzy = -(Vaa - V,,) cos 0 sin 0 sin 0
The following relations can be noted:
V. + Vyy +VzZ = 2Vaa +Vcc
V. -Vyy = (VI, -Vaa)sin2 z9(cos2 0-sin2 0)
(11)
Vyy -VzZ = -(V11 -Vaa)(cos2 0-sin2 t9sin2 0)
VzZ -V. = (V" -Vaa)(cos2 0-sin2 Ocos2 0)
Several distinct cases can be noted. In the first of these cases, when none of
the
cross-components is zero, VY # 0 nor Vyz # 0 nor Vz, ~ 0 then the azimuth
angle 0 is
not zero nor 7c/2 (90 ), and can be determined by,
0 =2tan-ly~'+yyx
(12) V. YY
0 = tan-1 Vyz = tan-1 V?v
Uxz vzx
By noting the relation,
(13) L2L = tan 0 sin 0 and V-Y = tan Ocos 0
Vxz Vyz
the dip (deviation) angle 0 is determined by,
2 2
(14) tan 0 = V~ + V'y
Vxz Vyz
In the second case, when V,.y = 0 and V yz = 0, then 0= 0 or 0= 0 or 7r (180 )
or
0 = 7r/2 (90 ) and 0 = 7r/2 (90 ), as the coaxial and the coplanar responses
should differ
from each other (Vaa#V,,).
If 0 = 0, then the dip angle 0 is determined by,
(15) 0=-~tan-l~xz+~zx
V. zz
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If 0 =7r (180 ), then the dip angle 0 is determined by,
(16) 0=+1tan-1Vxz+VZ"
2 V. - VzZ
Also, with regard to the second case, If 0 = 0, then V,, = Vyy and Vzx = 0. If
0 = 7r/2 (90 ) and 0 = n/2 (90 ), then Vz, = V, and Vzx = 0. These instances
are further
discussed below with relation to the fifth case.
In the third case, when V., = 0 and Vxz = 0, then 0_ n/2 (90 ) or 0= 0 or
0 = 0 and 0 = 7r/2 (90 ).
If 0 =7r/2, then the dip angle 0 is determined by,
(17) O= -1 tan-1 Vyz +VzY
2 Vyy - Vzz
If 0 =-7r/2, then the dip angle 0 is determined by,
(18) 0 _ + 1 tan-1 Vyz + Vzy
2 Vyy - Vzz
Also with regard to the third case, If 0 = 0, then V, = Vyy and Vyz = 0. If 0=
0
and0 = n/2 (90 ), Vyy = Vzz and Vyz = 0. These situations are further
discussed below with
relation to the fifth case.
In the fourth case, Vxz = 0 and Vyz = 0, then 0 = 0 or 7r(180 ) or 71/2 (90
).
If 0 = n/2, then the azimuth angle 0 is determined by,
(19) 0=-2tan-1~~'+~yx
V. yy
Also with regard to the fourth case, if 0 = 0 or 7r(180 ), then V, = Vyy and
Vyz = 0.
This situation is also shown below with relation to the fifth case.
In the fifth case, all cross components vanish, VxZ = VyZ = Vxy = 0, then 0 =
0, or
0 = n/2 (90 ) and 0 = 0 or n/2 (90 ).
If Vxx = Vyy then 0 = 0 or 7r(180 ).
If Vyy = Vzz then 0 = n/2 (90 ) and 0 = 0.
If Vzz = V, then 0 = n/2 (90 ) and 0 = 7r/2 (90 ).
Tool rotation around the tool/borehole axis
In the above analysis, all the transient responses Vz~(t) (i, j= x, y, z) are
specified by
the x-, y-, and z-axis directions of the tool coordinates. However, the tool
rotates inside the
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borehole and the azimuth orientation of the transmitter and the receiver no
longer coincides
with the x- or y-axis direction as shown in FIG. 5. If the measured responses
are
Vi~ (i , j= z), where z and y axis are the direction of antennas fixed to the
rotating
tool, and Vis the tool's rotation angle, then
Vzz Vzy Vzz V. VIy Vxz
(20) Vyz Vyy Vyz = R(V),r Vyx Vyy Vyz R(V)
vz.x vzy vzz vZx vzy vzz
cos sin V 0
(21) R(V) = sin V cos V 0
0 0 1
Then,
V z z =(Vaacos20 +V, sin2O)cos2(o -yz)+Vaasin2(0 -yz)
(22) Vyy =(Vaacos20 +V, sin29)sin2(0 -yz)+Vaacos2(0 -yz)
Vzz =Vaa sin2 a+V, COs2 a
Vzy = Vyz =-(Vaa - V, ) sin 2Ocos(o - V) sin(o - V)
(23) Vzz = Vzz = -(Vaa - V,,) cos Osin Ocos(o - V)
Vyz = Vzy = -(Vaa - V,,) cos Osin Osin(o - V)
The following relations apply:
Vzz + Vyy + VzZ = 2Vaa + Ucc
(24) Vzz -Vyy (Vcc -Vaa)sin24os2(~_)_sin2(~_)}
1//Vyy -VzZ =-(V11 -Vaa){cos2 0-sin2 Osin2(o -V)l
VzZ -Vzz = (V,, -Vaa){cos2 0-sin2 Ocos2(0 -01
Consequently,
j _I Vzy +Vyz
~-t/~= 2tan Vzz -Vyy
(25)
~ ~
tan-1 Vyz = tan-1 V?v
0 -
Vzz Vzz
The azimuth angle 0 is measured from the tri-axial responses if the tool
rotation
angle Vis known. To the contrary, the dip (deviation) angle 0 is determined by
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(26) tan 19 = xy
FiV_'_ Z 2 V ~ 2
Vyz
without knowing the tool orientation
Apparent dip angle and azimuth angle and the distance to the anomaly
The dip and the azimuth angle described above indicate the direction of a
resistivity
anomaly determined by a combination of tri-axial transient responses at a time
(t) when the
angles have deviated from a zero value. When t is small or close to zero, the
effect of such
anomaly is not apparent in the transient responses as all the cross-component
responses are
vanishing. To identify the anomaly and estimate not only its direction but
also the distance,
it is useful to define the apparent azimuth angle Opp(t) by,
_I Vxy (t) +Vyx (t)
~app (t) _ 2 tan Vxx (t) -Vyy (t)
(27)
Oapp (t) = tan-1 ~yz (t) = tan-1 ~z' (t)
xz zx
and the effective dip angle 0app(t) by
2 2
(28) tan 0app (t) V~ (t) + V (t)
xz yz
for the time interval when Oapp(t):#0 nor 7r,/2 (90 ). For simplicity, the
case examined below
is one in which none of the cross-component measurements is identically zero:
VY(t):#0,
Vyz(t):#0, and V,(t):#0.
For the time interval when Oapp(t) = 0, 0app(t) is defined by,
(29) 0 a p p (t) _ -1 tan-1 Vxz (t) +Vz, (t)
2 V-11 (t)-VzZ(t)
For the time interval when Oapp(t) = 7t/2 (90 ), 0app(t) is defined by,
(30) 0app (t) 2 1 tan yyz V ~
yy (t) zz (t)
When t is small and the transient responses do not see the effect of a
resistivity
anomaly at distance, the effective angles are identically zero, Opp(t)
=0app(t) = 0. As t
increases, when the transient responses see the effect of the anomaly, Oapp(t)
and 0app(t)
begin to show the true azimuth and the true dip angles. The distance to the
anomaly may be
indicated at the time when Opp(t) and 0app(t) start deviating from the initial
zero values. As
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shown below in a modeling example, the presence of an anomaly is detected much
earlier
in time in the effective angles than in the apparent conductivity (6pp(t)).
Even if the
resistivity of the anomaly may not be known until 6pp(t) is affected by the
anomaly, its
presence and the direction can be measured by the apparent angles. With
limitation in time
measurement, the distant anomaly may not be seen in the change of 6pp(t) but
is visible in
O,pp(t) and z9app(t).
First modeling Example
FIG. 6 depicts a simplified modeling example wherein a resistivity anomaly A
is
depicted in the form of, for example, a massive salt dome in a formation 5.
The salt
interface 55 may be regarded as a plane interface. FIG. 6 also indicates
coaxia160,
coplanar 62, and cross-component (64) measurement arrangements, wherein a
transmitter
coil and a receiver coil are spaced a distance L apart from each other. It
will be understood
that in a practical application, separate tools may be employed for each of
these
arrangements, or a multiple orthogonal tool. For further simplification, it
can be assumed
that the azimuth direction of the salt face as seen from the tool is known.
Accordingly, the
remaining unknowns are the first distance D 1 to the salt face 55 from the
tool, the second
distance D2 of the other side of the salt from the tool, the isotropic or
anisotropic formation
resistivity, and the approach angle (or dip angle) 0 as shown in FIG. 6. The
thickness A of
the salt dome is defined as A = D2 - D1. In case the resistivity in the
anomaly A is
anisotropic, the electromagnetic properties of the anomaly may be
characterized by normal
resistivity RL in the direction of the principal axis of the anisotropy (or
normal
conductivity (T1), and in-plane resistivity R// (or in-plane conductivity
(7//) in any direction
within a plane perpendicular to the principal axis. In case of anisotropy, R//
:# Rl.
Before discussing anisotropy in more detail, isotropic formations will first
be
illustrated with resistivity R (= R// = RL) (or its inverse 6= 6// = (T 1).
FIG. 7 and FIG. 8 show the calculated transient voltage response (V) from
coaxial
V,,(t) (line 65), coplanar V,(t) (line 66), and cross-component V,(t) (line
67)
measurements for a tool having L = 1 m, for 0 = 30 and located at a distance
of Dl = 10
m respectively Dl = 100 m away from a salt face 55. In the calculations, D2
has been
assumed much larger than 100 m, such that within the timescale of the
calculation (up to 1
sec) any influence from the other side of the salt A is not detectable in the
transient
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response. Moreover, when the anomaly is large and distant compared to the
transmitter-
receiver spacing L, the effect of the spacing L can be ignored and the
transient responses
can be approximated with those of the receivers near the transmitter.
The effect of the resistivity anomaly A (as depicted in FIG. 6) is seen in the
calculated transient responses as time increases. In addition to the coaxial
and coplanar
responses (65,66), the cross-component responses Vij(t) (i # j; I, j = x, y,
z) become non-
zero. In order to facilitate analysis of the responses, they may be converted
to apparent dip
and/or apparent conductivity.
The apparent dip angle Oapp(t), as calculated by
(31) 9 a p p (t) _ -1 tan-I Vz, (t) +Vxz (t)
2 VzZ (t) - V" (t)
is shown in FIG. 9 for a L = 1 m tool assembly when the salt face 55 is Dl =
10 m away
and at the approach angle of 0= 30
The apparent conductivity (6app(t)) from both the coaxial (Vzz(t) of FIG. 7)
and
the coplanar (Vxx(t) of FIG. 7) responses are shown in FIG. 10 (lines 68,
respectively line
69), wherein the approach angle (0 = 30 ) and salt face distance (Dl = 10 m)
are the same
as in FIG. 9. Details of how the apparent conductivities are calculated will
be provided
below.
Note that the true direction from the tool to the salt face (i.c. 30 ) is
reflected in the
apparent dip Oapp(t) plot of FIG. 9 as early as 10-4 second, when the presence
of the
resistivity anomaly is barely detected in the apparent conductivity (6pp(t))
plot of FIG. 10.
It takes almost 10-3 second for the apparent conductivity to approach an
asymptotic
6app(late t) value.
FIG. 11 shows the apparent dip Oapp(t) for the L = lm tool assembly when the
salt
face is D = lOm away, but at different angles between the tool axis and the
target varying
from 0 to 90 in 15 increments. The approach angle (0) may be reflected at
any angle in
about 10-4 sec.
FIG. 11 and FIGs. 12 and 13 compare the apparent dip 9pp(t) for different salt
face
distances (D = 10 m; 50 m; and 100 m) and different angles between the tool
axis and the
target.
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The distance to the salt face can be also determined by the transition time at
which
9,pp(t) takes an asymptotic value. Even if the salt face distance (D) is 100
m, it can be
identified and its direction can be measured by the apparent dip 9pp(t).
In summary, the method considers the coordinate transformation of transient EM
responses between tool-fixed coordinates and anomaly-fixed coordinates. When
the
anomaly is large and far away compared to the transmitter-receiver spacing,
one may
ignore the effect of spacing and approximate the transient EM responses with
those of the
receivers near the transmitter. Then, one may assume axial symmetry exists
with respect to
the c-axis that defines the direction from the transmitter to the anomaly. In
such an axially
symmetric configuration, the cross-component responses in the anomaly-fixed
coordinates
are identically zero. With this assumption, a general method is provided for
determining
the direction to the resistivity anomaly using tri-axial transient EM
responses.
The method defines the apparent dip 9pp(t) and the apparent azimuth (Oapp(t)
by
combinations of tri-axial transient measurements. The apparent direction {
9pp(t), (Opp(t)}
reads the true direction 10, (o} at later time. The 9pp(t) and (opp(t) both
read zero when t is
small and the effect of the anomaly is not sensed in the transient responses
or the apparent
conductivity. The conductivities (6coaxzal(t) and 6copiawr(t)) from the
coaxial and coplanar
measurements both indicate the conductivity of the near formation around the
tool.
Deviation of the apparent direction ((9app(t), (Oapp(t))) from zero identifies
the
anomaly. The distance to the anomaly is measured by the time when the apparent
direction
(I Oapp(t), (oapp(t))) starts to deviate from zero or by the time when the
apparent direction
(I Oapp(t), (oapp(t))) starts approaches the true direction ((9, 0). The
distance can be also
measured from the change in the apparent conductivity. However, the anomaly is
identified
and measured much earlier in time in the apparent direction than in the
apparent
conductivity.
Apparent ConductivitX
As set forth above, apparent conductivity can be used as an alternative
technique to
apparent angles in order to determine the location of an anomaly in a
wellbore. The time-
dependent apparent conductivity can be defined at each point of a time series
at each
logging depth. The apparent conductivity at a logging depth z is defined as
the conductivity
of a homogeneous formation that would generate the same tool response measured
at the
selected position. In transient EM logging, transient data are collected at a
logging
depth or tool location z as a time series of induced voltages in a receiver
loop. Accordingly,
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time dependent apparent conductivity (6(z; t)) may be defined at each point of
the time
series at each logging depth, for a proper range of time intervals depending
on the
formation conductivity and the tool specifications.
The induced voltage of a coaxial tool with transmitter-receiver spacing L in
the
homogeneous formation of conductivity (6) is given by,
/
z
(32) Vzz (t) = C ( g ~ e-u
2
where u 2= u 6 L and C is a constant.
4 t
The time-changing apparent conductivity depends on the voltage response in a
coaxial tool (Vzz(t)) at each time of measurement as:
(33) C g t/ I,u 6 (t)1 ` e - -uPP a (t)~ - V~ (t)
L2
where uapp (t) 2=Po 6app (t) and Vzz(t) on the right hand side is the measured
voltage
4 t
response of the coaxial tool. From a single type of measurement (coaxial,
single spacing),
the greater the spacing L, the larger the measurement time (t) should be to
apply the
apparent conductivity concept. The 6pp(t) should be constant and equal to the
formation
conductivity in a homogeneous formation: 6pp(t) = 6 The deviation from a
constant (6) at
time (t) suggests a conductivity anomaly in the region specified by time (t).
The induced voltage of the coplanar tool with transmitter-receiver spacing L
in the
homogeneous formation of conductivity (6) is given by,
(34) V~(t)=CWgt 6Y2,)' (1-u2 )e-"z
where u 2='u 6 LZ and C is a constant. At small values of t, the coplanar
voltage changes
4t
polarity depending on the spacing L and the formation conductivity.
Similarly to the coaxial tool response, the time-changing apparent
conductivity is
defined from the coplanar tool response Vx(t) at each time of measurement as,
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)/
(35) ~' (uo 6ap (t) 2 (1 uapp (t) 2 )eul(t)"
UxX (O
8t 2
2
where uapp (t) 2 =~0 6app (t) L and Vx(t) on the right hand side is the
measured voltage
4 t
response of the coplanar tool. The longer the spacing, the larger the value t
should be to
apply the apparent conductivity concept from a single type of measurement
(coplanar,
single spacing). The 6pp(t) should be constant and equal to the formation
conductivity in a
homogeneous formation: 6pp(t) = 6
When there are two coaxial receivers, the ratio between the pair of voltage
measurements is given by,
(36) V~(Ll;t) =e- 4t(L,z -Lzz )
Vzz (L2 ; t)
where LI and L2 are transmitter-receiver spacing of two coaxial tools.
Conversely, the time-changing apparent conductivity is defined for a pair of
coaxial
tools by,
-ln Vzz(L';t)
Vzz (Lz ; t) 4t
(37) 6aPP (t) = z z
L1 - LZ uo
at each time of measurement. The 6pp(t) should be constant and equal to the
formation
conductivity in a homogeneous formation: 6pp(t) = 6
The apparent conductivity is similarly defined for a pair of coplanar tools or
for a
pair of coaxial and coplanar tools. The 6pp(t) should be constant and equal to
the
formation conductivity in a homogeneous formation: 6pp(t) = 6 The deviation
from a
constant (6) at time (t) suggests a conductivity anomaly in the region
specified by time (t).
As will be illustrated below, apparent conductivity (6pp(t)), whether coaxial
or
coplanar, may reveal three parameters in relation to a two-layer formation,
including:
(1) the conductivity of a local first layer in which the tool is located;
(2) the conductivity of one or more adjacent layers or beds; and
(3) the distance of the tool to the layer boundaries.
Analysis of Coaxial Transient Response in Two-layer Models
To illustrate usefulness of the concept of apparent conductivity, the
transient
response of a tool in a two-layer earth model, as in FIG. 14 for example, can
be examined.
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FIG. 14 illustrates a coaxial too180 in which both a transmitter coil (T) and
a
receiver coil (R) are wound around the common tool axis z and spaced a
distance L apart.
The symbols 61 and 62 may represent the conductivities of two formation
layers. The
coaxial too180 be placed in a horizontal we1188 traversing formation layer 5
and extending
parallel to the layer interface 55.
In the present example, a horizontal well is depicted such that the distance
from the
tool to the layer boundary corresponds to the distance of the horizontal
borehole to the
layer boundary. Under a more general circumstance, the relative direction of a
borehole
and tool to the bed interface is not known.
The calculated transient voltage response V(t) for the L = 1 m transmitter-
receiver
offset coaxial tool at various distances D between the too180 and the layer
boundary 55 is
shown in FIG. 15 for D = 1, 5, 10, 25, and 50 m. The formation can be analyzed
using
these responses, employing apparent conductivity as further explained with
regard to FIGs.
16 and 17.
FIG. 16 shows the voltage data of FIG. 15 plotted in terms of apparent
conductivity, for a geometry wherein 61 = 0.1 S/m (RI = 10 S2m) and 62 = 1 S/m
(R2 = 1
S2m). Similarly, FIG. 17 illustrates the apparent conductivity in a two-layer
model where
61 = 1 S/m (RI = 1 S2m) and 62 = 0.1 S/m (R2 = 10 S2m).
The apparent conductivity plots reveal a "constant" conductivity at small t,
and at
large t but having a different value, and a transition time tc that marks the
transition
between the two "constant" conductivity values and depends on the distance D.
As will be further explained below, in a two-layer resistivity profile, the
apparent
conductivity as t approaches zero can identify the layer conductivity 61
around the tool,
while the apparent conductivity as t approaches infinity can be used to
determine the
conductivity 62 of the adjacent layer at a distance. The distance to the bed
boundary 55
from the too180 can also be measured from the transition time tc observed in
the apparent
conductivity plots.
At small values of t, the tool reads the apparent conductivity 61 of the first
layer 5
around the too180. Conductivity at small values of t is thought to correspond
to the
conductivity of the local layer 5 where the tool is located in. At small
values of t, the signal
reaches the receiver directly from the transmitter without interfering with
the bed
boundary. Namely, the signal is affected only by the conductivity 61 around
the tool.
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At large values of t, the tool reads 0.4 S/m for a two-layer model where
either 61=
1 S/m (Ri =1 S2m) and 62 = 0.1 S/m (R2 = 10 S2m), or 6i = 0.1 S/m (Ri = 10
S2m) and 62 =
1 S/m (R2 = 1 S2m). The value of 0.4 is believed to correspond to some average
between
the conductivities of the two layers, because at large values of t, nearly
half of the signals
come from the formation below the tool and the remaining signals come from
above, if the
time for the signal to travel the distance between the tool and the bed
boundary is small.
This is further investigated in FIG. 18, which shows examples of the 6pp(t)
plots
for D = 1 m and L=1 m, but for different resistivity ratios of the target
layer 2 while the
local conductivity (61) is fixed at 1 S/m (Ri =1 S2m). The apparent
conductivity at large
values of t is determined by the target layer 2 conductivity, as shown in line
71 in FIG. 19
when 6i is fixed at 1 S/m.
Numerically, the late time conductivity may be approximated by the square root
average of two-layer conductivities as:
V61 + 62
(39) UaPP (t _> ; 61 , 62 ) _ .
2
This is depicted as line 72 in FIG. 19.
Thus, the conductivity at large values of t (as t approaches infinity) can be
used to
estimate the conductivity (62) of the adjacent layer when the local
conductivity (6i) near
the tool is known, for instance from the conductivity as t approaches zero as
illustrated in
FIG. 20.
Estimation of D, The Distance to the electromagnetic anomaly
The distance D from the tool to the bed is reflected in the transition time
tc. The
transition time at which the apparent conductivity ((Tapp(t)) starts deviating
from the local
conductivity ((T1) towards the conductivity at large values of t depends on D
and L, as
shown in FIG. 21.
For convenience, the transition time (tc) can be defined as the time at which
the
6app(tc) takes a cutoff conductivity ((Tc). In this case, the cutoff
conductivity is
represented by the arithmetic average between the conductivity as t approaches
zero and
the conductivity as t approaches infinity. The transition time (tJ is dictated
by the ray path
RP:
(40) RP= ~~~~+
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that is the shortest distance for the electromagnetic signal traveling from
the transmitter to
the bed boundary, to the receiver, independently of the resistivity of the two
layers.
Conversely, the distance (D) to the anomaly can be estimated from the
transition time (tJ,
as shown in FIG. 22.
Look-Ahead Capabilities of EM Transient Method
By analyzing apparent conductivity or its inherent inverse equivalent
(apparent
resistivity), the present invention can identify the location of a resistivity
anomaly (e.g., a
conductive anomaly and a resistive anomaly). Further, resistivity or
conductivity can be
determined from the coaxial and/or coplanar transient responses. As explained
above, the
direction to the anomaly can be determined if the cross-component data are
also available.
To further illustrate the usefulness of these concepts, the foregoing analysis
may also be
used to detect an anomaly at a distance ahead of the drill bit.
FIG. 23 shows a coaxial tool 80 with transmitter-receiver spacing L placed in,
for
example, a vertical we1188 approaching or just beyond an adjacent bed that is
a resistivity
anomaly. The too180 includes both a transmitter coil T and a receiver coil R,
which are
wound around a common tool axis and are oriented in the tool axis direction.
The symbols
61 and 62 may represent the conductivities of two formation layers, and D the
distance
between the too180 (e.g. the transition antenna T) and the layer boundary 55.
The calculated transient voltage response of the L=1 m (transmitter-receiver
offset) coaxial tool at different distances (D = 1, 5, 10, 25, and 50 m) as a
function of t is
shown in FIG. 24, in a case wherin 61 = 0.1 S/m (corresponding to RI = 10
S2m), and 62
= 1 S/m (corresponding to R2 = 1 S2m). Though difference is observed among
responses at
different distances, it is not straightforward to identify the resistivity
anomaly directly from
these responses.
The same voltage data of FIG. 24 is plotted in terms of the apparent
conductivity
(6app(t)) in FIG. 25. From this Figure, it is clear that the coaxial response
can identify an
adjacent bed of higher conductivity at a distance. Even a L = 1 m tool can
detect the bed at
10, 25, and 50 m away, if low voltage response can be measured for 0.1-1
seconds long.
The 6pp(t) plot exhibits at least three parameters very distinctly in the
figure: the
early time conductivity; the later time conductivity; and the transition time
that moves as
the distance (D) changes. In FIG. 25, it should be noted that, at early time
whereby t is
close to zero, the tool reads the apparent conductivity of 0.1 S/m, which is
representative of
the layer just around the tool. The signal that reaches the receiver R not yet
contains
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information about the boundary 55. At later time, the tool reads close to 0.55
S/m,
representing an arithmetic average between the conductivities of the two
layers. At later
time, t--> oo, nearly half of the signals come from the formation below the
tool and the
other half from above the tool, if the time to travel the distance (D) of the
tool to the bed
boundary is small. The distance D is reflected in the transition time tc.
FIG. 26 illustrates the 6app(t) plot of the coaxial transient response in the
two-layer
model of FIG. 23 for an L = 1 m tool at different distances (D), except that
the conductivity
of the local layer ((T1) is 1 S/m (R1 = 1 S2m) and the conductivity of the
target layer ((72) is
0.1 S/m (R2 = 10 S2m). Again, the tool reads at early time the apparent
conductivity of 1.0
S/m that is of the layer just around the tool. At a later time, the tool reads
about 0.55 S/m,
the same average conductivity value as in FIG. 25. The distance (D) is
reflected in the
transition time tc.
Hence, the transient electromagnetic response method can be used as a look-
ahead
resistivity logging method.
FIG. 27 compares the 6app(t) plot of FIG. 25 and FIG. 26 for L = 1 m and D
50 m. The late time conductivity is determined solely by the conductivities of
the two
layers (61 and (72) alone. It is not affected by where the tool is located in
the two layers.
However, because of the deep depth of investigation, the late time
conductivity is not
readily reached even at t = 1 second, as shown in FIG. 25. In practice, the
late time
conductivity may have to be approximated by 6app(t = 1 second) which slightly
depends
on D as illustrated in FIG. 25.
Numerically, the late time apparent conductivity may be approximated by the
arithmetic average of two-layer conductivities as: 6app (t 61, 62 )= 61 + 62 .
This is
2
reasonable considering that, with the coaxial tool, the axial transmitter
induces the eddy
current parallel to the bed boundary. At later time, the axial receiver
receives horizontal
current nearly equally from both layers. As a result, the late time
conductivity must see
conductivity of both formations with nearly equal weight.
FIG. 28 compares the 6pp(t) plots for D = 50 m but with different spacing L.
The
6,pp(t) reaches a nearly constant late time apparent conductivity at later
times as L
increases. The late time apparent conductivity (6pp(t--> oo) is nearly
independent of L.
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However, the late time conductivity defined at t = 1 second, depends on
slightly the
distance (D).
Thus, the late time apparent conductivity (6pp(t--> oo)) at t = 1 second can
be used
to estimate the conductivity of the adjacent layer (62) when the local
conductivity near the
tool (61) is known, for instance, from the early time apparent conductivity
(6pp(t-->0) _
61).
Estimation of the distance (D) to the electromagnetic anomaly
The transition time (tc) at which the apparent conductivity starts deviating
from the
local conductivity (61) toward the late time conductivity clearly depends on
D, the distance
of the tool to the bed boundary, as shown in FIG. 25 for a L = 1 m tool.
For convenience, the transition time (tc) is defined by the time at which the
6pp(tj
takes the cutoff conductivity (6,), that is, in this example, the arithmetic
average between
the early time and the late time conductivities: 6c =(6pp(t--;0)+6pp(t---;~-
)J/2. The
transition time (tc) is dictated by the ray-path RP, D minus L/2 that is, half
the distance for
the EM signal to travel from the transmitter to the bed boundary to the
receiver,
independently on the resistivity of the two layers. Conversely, the distance
(D) can be
estimated from the transition time (tc), as shown in FIG. 29 when L = 1 m.
Analysis of Coplanar Transient Responses in Two-Layer Models
While the coaxial transient data were examined above, the coplanar transient
data
are equally useful as a look-ahead resistivity logging method.
FIG. 30 shows a coplanar too180 with transmitter-receiver spacing L placed in
a
we1188 and approaching (or just beyond) layer boundary 55 of an adjacent bed
that is the
resistivity anomaly. On the coplanar tool, both a transmitter T and a receiver
R are oriented
perpendicularly to the tool axis z and parallel to each other. The symbols 61
and 62 may
represent the conductivities of two formation layers.
Corresponding to FIG. 25 for coaxial tool responses where L = 1 m, the
apparent
conductivity (6pp(t)) for calculated coplanar responses is plotted in FIG. 31
for different
tool distances from the bed boundary 55. It is clear that the coplanar
response can also
identify an adjacent bed of higher conductivity at a distance. Even a L = 1 m
tool can
detect the bed at 10 m, 25 m, and 50 m away if low voltage responses can be
measured for
0.1-1 seconds long. The 6pp(t) plot for the coplanar responses exhibits three
parameters
equally as well as for the coaxial responses.
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Like it was the case in the co-axial geometry, it is also true for the
coplanar
responses that the early time apparent conductivity (6pp(t-->0)) is the
conductivity of the
local layer (61) where the tool is located. Conversely, the layer conductivity
can be
measured easily by the apparent conductivity at earlier times.
The late time apparent conductivity (6pp (t--> oo)) is some average of
conductivities
of both layers. The conclusions derived for the coaxial responses apply
equally well to the
coplanar responses. However, the value of the late time conductivity for the
coplanar
responses is not the same as for the coaxial responses. For coaxial responses,
the late time
conductivity is close to the arithmetic average of two-layer conductivities in
two-layer
models.
FIG. 32 shows the late time conductivity (6pp(t-->oo)) for coplanar responses
as
obtained from the model calculations (line 77) whereby D = 50 m and L = 1 m,
but for
different conductivities of the local layer while the target conductivity is
fixed at 1 S/m.
Late time conductivity is determined by the local layer conductivity, and is
numerically
I + 6 2
close to the square root average as, U aPP (t 6 1 , 6 2 2
as is shown by line 78 in FIG. 32.
To summarize, the late time conductivity
((Tapp(t--> oo)) can be used to estimate the conductivity of the adjacent
layer ((72) when the
local conductivity near the tool ((T1) is known, for instance, from the early
time
conductivity (6app(t-->0) = (71). This is illustrated in FIG. 33 wherein line
79 has been
obtained from model calculations and line 79a displays the average
approximation.
Estimation of the distance (D) to the electromagnetic anomaly
The transition time tc at which the apparent conductivity starts deviating
from the
local conductivity (6i) toward the late time conductivity clearly depends on
the distance
(D) of the too180 (e.g. the transmitter T) to the bed boundary 55, as shown in
FIG. 30.
The transition time (tJ may be defined by the time at which the 6pp(tj takes
the
cutoff conductivity (6,) that is, in this example, the arithmetic average
between the early
time and the late time conductivities: 6, =(6pp(t--;0)+6pp(t---;~-) J/2. The
transition time
(tJ is dictated by the ray-path, D minus L/2 that is, half the distance for
the EM signal to
travel from the transmitter to the bed boundary to the receiver, independently
of the
resistivity of the two layers.
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Conversely, the distance (D) can be estimated from the transition time (tj, as
shown in FIG. 34, where L = 1 m.
Analysis of transient electromagnetic response data for three or more
formation layers
The next model shows a conductive near layer, a very resistive layer, and a
further
conductive layer. The geological configuration is depicted in FIG. 35,
together with a
coaxial too180 in a relatively conductive formation 82 wherein an anomaly is
located in
the form of a relatively resistive layer 83. As shown, the formation on the
other side of
layer 83, as seen from too180 and identified in FIG. 35 by reference
numera184, is
identical to the formation 82 on the tool side of the layer 83. However, the
method will also
work if the formation 84 on the other side of layer 83 would constitute a
layer that has
different properties from those of the near formation 82.
In either case, the tool "sees" the anomaly 83 as a first layer at a first
distance D1
away and having a thickness A, and it "sees" the formation on the other side
of the
anomaly 83 as a second layer 84 at a second distance D2 = D1 + A away and
having
infinite thickness.
FIG. 36 is a graph showing calculated apparent resistivity response Rapp
versus time t for
a geometry as given in FIG. 35. For the calculation of FIG. 36, it has been
assumed that the
anomaly is formed of a resistive salt bed, having a resistivity of 100 S2m,
and that the
formation is formed of for instance a brine-saturated formation having a
resistivity of 1
S2m. The tool has been modeled as being oriented with its main axis parallel
to the first
interface 81 between the brine-saturated formation 82, and the distance
between the main
axis and the first layer 83, D1, has been taken 10 m. The resistive bed
thickness A has been
varied from a fraction of a to 100 meters in thickness.
The first climb of Rapp(t) is the response to the salt and takes place at 104
s with an
L=1 m tool when the salt is at Dl = 10 m away. If the salt is fully resolved
(by infinitely
thick salt beyond D 1=10 m), the apparent resistivity should read 3 S2m
asymptotically. The
subsequent decline of Rapp(t) is the response to a conductive formation behind
the salt
(resistive bed). Rpp(late t) is a function of conductive bed resistivity and
salt thickness. If
the time measurement is limited to 10-2 s, the decline of Rpp(t) may not be
detected for the
salt thicker than 500 m.
With respect to the resistive bed resolution, the coaxial responds to a thin
(1-2m
thick) bed. The time at which Rpp(t) peaks or begins declining depends on the
distance to
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the conductive bed behind the salt. As noted previously, when plotted in terms
of apparent
conductivity 6pp(t), the transition time may be used to determine the distance
to the
boundary beds.
Another three-layer formation was also modeled, as shown in FIG. 37. In this
instance, the intermediate layer 83 was a more conductive layer than the
surrounding
formation 82. This conductive bed 83 may be considered representative of, for
instance, a
shale layer. The coaxial too180, having an L = 1 m spacing, is located in a
borehole in a
formation 82 having a resistivity of 10 S2m and is located D1 = 10 m from the
less resistive
(more conductive) layer 83, which has a resistivity of 1 S2m. The third layer
84 is beyond
the conductive bed 83 and has a resistivity of 10 S2m as does layer 82. The
conductive bed
83 was modeled for a range of thicknesses A varying from fractions of a meter
up to an
infinite thickness. The apparent resistivity, as calculated, is set forth in
FIG. 38.
The decrease in Rpp(t), which can be seen in FIG. 38, is attributed to the
presence
of the shale (conductive) layer and appears as t--> 10-5 s. The shale response
is fully
resolved by an infinitely thick conductive layer that approaches 3 S2m. The
subsequent rise
in Rpp(t) is in response to the resistive formation 84 beyond the shale layer
83. The
transition time is utilized to determine the distance D2 from the too180 to
the interface 85
between the second and third layers (83 respectively 84). Rpp(late t) is a
function of
conductive bed resistivity. As the conductive bed thickness A increases, the
time
measurement must likewise be increased (> 10-2s) in order to measure the rise
of RIPp(t)
for conductive layers thicker than 100 m.
Still another three-layer model is set forth in FIG. 39, wherein the coaxial
too180 is
in a conductive formation 82 (1 S2m), and a highly resistive second layer 84
(100 S2m) as
might be found in, for instance, a salt dome. Formation 82 and the second
layer 84 are
separated by a first layer 83 that has an intermediate resistance (10 S2m).
The thickness A
has been varied in the calculations of the apparent resistivity response, as
depicted in FIG.
40.
The response to the intermediate resistive layer is seen at 10-4 s, where
RIPp(t)
increases. If the first layer 83 is fully resolved by an infinitely thick bed,
the apparent
resistivity approaches a 2.6 S2m asymptote. As noted in FIG. 40, the Rpp(t)
undergoes a
second stage increase in response to the 100 S2m highly resistive second layer
84. Based on
the transition time, the distance to the interface is determined to be 110 m.
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Though complex, the apparent resistivity or apparent conductivity in the above
examples delineates the presence of multiple layers. The observed changes of
apparent
conductivity (or apparent resistivity) allow determination of the distances D1
and D2.
Transient electromagnetic responses involving formation anisotropy
As stated above, an electromagnetic anomaly may display anisotropic
electromagnetic properties. An example is shown in FIG. 6, if R// # Rl.
Various mechanisms may give rise to a macroscopic electromagnetic induction
effect. For instance, oriented fractions may generate an anisotropic response.
Electromagnetic anisotropy may also arise intrinsically in certain types of
formations, such
as shales, or it may arise as a result of sequences of relatively thin layers.
In the way as depicted in FIG. 6, the principal anisotropy direction
corresponds to
the approach angle 0. This correspondence is mainly for reasons of simplicity
in setting
forth the embodiments, and need not necessarily be the case in every situation
within the
scope of the invention.
In the following it will be explained how electromagnetic anisotropy of at
least one
of the formation layers may be taken into account when analyzing time-
dependent transient
response signals. This may comprise determining one or more anisotropy
parameters that
characterize the anisotropic electromagnetic properties. Amongst anisotropy
parameters are
anisotropy ratio a2, anisotropic factor 0, conductivity along a principal
anisotropy axis 61
(or resistivity along the principal anisotropy axis RL), conductivity in a
plane
perpendicular to the principal anisotropy axis 6// (or resistivity in a plane
perpendicular to
the principal anisotropy axis R//); tool axis angle relative to the principal
anisotropy axis.
Using the concepts of apparent conductivity or apparent resistivity and/or
apparent
dip or azimuth, the distance and/or direction to an anomaly may be determined
from the
time-dependent transient response signals even when the anomaly, and/or a
distant
formation layer, comprise(s) an electromagnetic anisotropy or when the
transmitter and/or
receiver antennae are embedded in an anisotropic formation layer.
Using the principles set forth above, the analysis taking into account
anisotropy
may be extended to multiple bedded formations, including those where only a
distant
formation layer or target anomaly gives anisotropic electromagnetic induction
responses
(such as for instance in FIG. 6) or where a local formation layer wherein the
transmitter
and receiver antennae are located, displays anisotropic behavior and one or
more other,
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isotropic or anisotropic layers are present at a distance. The distance and
direction from the
tool to the more distant layers and/or the target anomaly may then be
determined, provided
that anisotropy is taken into account.
In the forthcoming explanation, for reasons of simplicity, it will be assumed
that the
anisotropy has a vertically aligned principal axis, such that the angle
between the tool axis
z and the principal anisotropy axis corresponds to the dip angle or deviation
angle 0. The
term horizontal resistivity RH may be employed, which generally corresponds to
the
resistivity in the anisotropy plane perpendicular to the principal anisotropy
direction. The
term vertical resistivity RU generally refers to resistivity in the principal
anisotropy
direction or normal direction.
Transient EM responses in a homogeneous anisotropic formation
Considered is an anisotropic formation, in which a vertical resistivity RU (or
its
inverse vertical conductivity o-V) is different from the horizontal
resistivity RH (or
horizontal conductivity (TH). Assumed is that the formation is azimuth-
symmetric, in the
horizontal direction. The tool axis z is deviated from the vertical direction
by the dip
(deviation) angle 0 in the zx-plane. The transmitter antenna is placed at
origin. The
receiver antenna is placed at (x=L=sin6, y=O, z=L=cos6). There may be four
independent
combinations of transmitter and receiver orientations that render non-zero
responses.
In addition to a coaxial response, VZz, there are two coplanar responses, VXx
and
Vyy, and one cross-component response VXz = VZx= One coplanar response, VXx,
is
from a transverse transmitter antenna and receiver antenna that are oriented
within the zx-
plane. Another coplanar response, Vyy, is from a transverse transmitter and
receiver both
of which are oriented in the y-axis direction. The cross-component response is
from a
transverse receiver antenna with the longitudinally oriented transmitter
antenna, or vise
versa. The transverse receiver antenna is directed within the zx-plane. Any
cross-
component involving either a transmitter or a receiver oriented in the y-axis
direction, i.e.
Vyx and Vxy and Vyz and VZy are all vanishing.
The above has been set forth in tool-coordinates. It is further remarked that
any
antenna that is sensitive to a transverse component of an electromagnetic
induction field
suffices as a transverse antenna.
Applicants have derived the transient response in time domain, expressed in
terms
of horizontal conductivity 6H and anisotropic factor 0, are given by:
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(41) V z z (t) = c e u {l+ 2 (6 2 e u 00 1 ) -1) 4u2 (e u
8t~
)% z
~~+ cos B 1 W (~ze-u~(~~ ~) -1)- (e
xx (
(42)V (t) = Cuo~H e u
~1- sin B
8t~ ~ ~2 4u2
(43) V, (t)=VXz(t)=CBte u
{:::~2(~2e uz ; and
~
(44)
[1_u21_ 1 r1 ('82e-u2' (,8 2'-1)-1)- 1 (e-u~(82, -1)-1)
Vyy(t)=C(,u06HP e-u2, sin2B 2 4u2
8t % +~[a2(3-2u2~2)e-uz (lj z -1)-(3-2u2)1
In these equations, u2 ='u 6H LZ and C is a constant. The anisotropic factor
is defined
4 t
as:
(45) 8 = 1+(a2 -1)sin2 B; a2 = 6V
6g
The following remarks may be made based on these equations:
1. The coaxial response depends only on the horizontal resistivity RH(= 1/6H)
and the
anisotropic factor 0 that is determined by the anisotropy ratio
a2=6V/6H=RH/RV, and the
dip angle 0. Conversely, neither the anisotropy nor the dip angle can be
determined from
coaxial measurements alone.
2. Both coplanar responses depend on the horizontal resistivity, the
anisotropic factor,
and the dip angle.
3. In vertical boreholes with 0=0, the coaxial response depends only on the
horizontal
resistivity, while the coplanar reponse is determined by both the horizontal
resistivity and
the vertical resistivity.
4. In horizontal logging with 6=7r/2, the coaxial response depends on both the
horizontal resistivity and the vertical resistivity, but the coplanar response
is determined
solely by the horizontal resistivity.
5. Because u2-->0 as t--large, the dip angle is determined by:
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(46) 2V X, (t) = tan 20 + O(u 2), whereby O(u2) denotes a remainder on the
order of
V" (t) -Vzz (t)
u2.
Late time responses in a homogeneous anisotropic formation
Similar to the investigation set forth above with regard to layer models, the
late
time limits may be derived. As t--> -, u2-->0, and therefore these limits
converge. Taking
into account anisotropy, the late time limits of equations (41) to (44) are:
(47) Vzz (t) = C (U8t~ ), {1+(a2 -1)sin2B1 ;
(48) V,,(t)=C(Ugt~), ~1+~(a2 -1)cos2B
(49) V,, (t) = C gtY ~ 3 4(a2 -1)cos B sin B~ and
(50) VYy (t) = C 8tY {i+(a2_i)}.
The dip (deviation) angle is determined by:
(51) B = 1 tan-i 2V,, (t)
2 V" (t) -Vzz (t)
The anisotropy ratio a2 may be determined from:
V,(t+V~,(tl+g(a2 -1)
(52)
2VI,y(t-->1+4(az-1)
When the dip angle 0 is known or estimated, the anisotropy ratio may
alternatively
be detennined from:
3 (VXX (tVzz(t~~) - 4a~-1)
(53) V~ (t ~ +Vzz (t ~ 2 + 3 ~a~ -1) cos 2B
4
It is further remarked that the sum of the co-axial response with the Xx
coplanar
response is independent from the approach angle.
Apparent conductivity for co-axial and co-planar responses in a homogeneous
anisotropic
formation
Similar to the investigation set forth above with regard to layer models,
apparent
conductivity is also a useful derived formation quantity in case of an
anisotropic formation
layer.
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The apparent conductivity is defined for both coaxial ((TZZ(t)) and coplanar
(a-Xx(t), 6yy(t)) responses. The apparent conductivity is the time-varying
conductivity that
would give the measured coaxial or coplanar response at time t if the
formation would be
homogeneous and isotropic.
As before, the time-changing apparent conductivities depend on the voltage
response in a coaxial tool (Vz(t)) or in a coplanar tool (Vx,(t) at each time
of measurement
as:
(54) VZZ (t) = C ~uo6zZ (t)~/ e ~z
8t~ z
(55) Vx, (t)=C gt y"t)~ (1-u2 )e-"'
wherein
(56a) u2 = 'uo6zz (t) or
4t
(56b) u 2 = 'u 6Xx (t) LZ , respectively.
4t
Then, at large t, the apparent conductivity approaches the value determined by
the
anisotropic conductivity and the dip angle as follows:
~
(57) 6~ (t large) = 6H ~I+ ~(az -I)sin291 for coaxial response;
/
(58) 6~ (t large) = 6H ~I+ ~(az -I)cos29~ for Xx-coplanar response;
(59) 6Yy (t large) = 6H ~1+ 4 3(a~ -Ifor Yy-coplanar response.
In terms of the apparent conductivity,
(60) 6x, (t) + 6zz (t) ", = 6H Y {2+_(az -1; and
Llarge e 4
(61) 6x, (t)~ -6zz(t)", = 6Hy,, {_(az -1)cos20
~
Llarge e 4
The anisotropy ratio a2 may be estimated from the ratio of equations (61) and
(60),
and the estimated 0 as:
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6~(t)~ -6zz(t)~ _ (a2 -1~
(62) 4 cos 20
+ 6 2 z (t)~ t'large 2+ 4(Gk'2 -1)
Modeling examples
FIGs. 41 to 45 relate to transient electromagnetic induction measurements, and
analysis thereof, in a homogeneous anisotropic formation for various 02 (in
order of
increasing anisotropy: 1.0; 0.8; 0.6; 0.4; 0.3) for a coaxial L = 1 m tool.
Of these Figures, FIG. 41 shows the calculated coaxial voltage responses for a
formation wherein the conductivity in horizontal direction 6H = 1 S/m (RH = 1
S2m). The
lines show the voltage response as a function of time t (ranging from 1E-08
sec to 1E+00
sec on a logarithmic scale) after a step-wise sudden switching off of the
transmitter. Line
101 corresponds to a homogeneous isotropic formation ((32 = 1.0) and should
ideally
correspond to a dipole solution. Lines 102, 103, 104, and 105 represent
increasing
anisotropy and respectively correspond to (32 = 0.8, (32 = 0.6, (32 = 0.4, and
(32 = 0.3.
FIG. 42 shows the apparent conductivity that has been calculated from the
responses as shown in FIG. 41. The same line numbers have been used as in FIG.
41.
FIG. 43 is similar to FIG. 42 but it shows the apparent conductivity that has
been
derived from responses calculated for formations with 6H = 0.1 S/m (RH = 10
S2m). The
same general behavior is found.
FIG. 44 is similar to FIGs. 42 and 43, but it shows the apparent conductivity
that
has been derived from responses calculated for formations with 0-H = 0.01 S/m
(RH = 100
S2m). The same general behavior is again found.
In each of FIGs. 42, 43, and 44, the late time apparent conductivity is
constant for
each of the anisotropic factors, indicative of a macroscopically homogeneous
formation.
The late time apparent conductivity decreases with anisotropic factor as is
expected
because the vertical conductivity, along the principal axis of the anisotropy,
is lower than
the horizontal conductivity.
FIG. 45 plots the late time asymptotic value of coaxial apparent conductivity
2/3
6Zz(t~-) over 6H against ~1+ ~(,6z -1)f . The resulting straight line
demonstrates the
linear relationship. When taking into account the anisotropy, the correct
value of the
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horizontal formation resistivity (or conductivity) can thus be extracted from
the asymptotic
coaxial apparent conductivity values.
Even for highly anistropic formations, the apparent conductivity is almost
indistinguishable from apparent conductivity of a homogeneous isotropic
formation with a
lower conductivity. Interpretation mistakes may thus easily be made if
anisotropy is not
taken into account when analyzing.
As follows from the above, anisotropy can be taken into account, for instance
by
combining co-axial responses with coplanar responses. The precise embodiment
depends
on which of the parameters are known or estimated. The sum of the co-axial
response with
the Xx coplanar response is independent from the approach angle. If C and 6H
are known
or estimated then the anisotropy ratio a2 follows from the late time value of
sum VZz +
VXx. If, on the other hand, the approach angle 0 is known, C and 6H don't need
to be
known because the anisotropy ratio a2 may be derived from Eq. (53). If none of
the other
parameters is known, Eq. (52) may be employed requiring combining co-axial
response
with two independent co-planar responses.
Apparent dip in a homogeneous anisotropic formation
In FIG. 46, apparent dip angles Oapp(t) derived using Eq. (51) from calculated
coaxial, coplanar and cross-component transient responses from a L = 1 m tool
in a
formation of RH = 10 S2m and RV/RH = 9, for various approach angles, or dip
angles.
Line 106 corresponds to 6= 30 ; line 107 to 6= 45 ; line 108 to 6= 60 ; and
line 109 to 0
= 75 .
The dip angle is thus reflected accurately by the asymptotic value of the
apparent dip. The
asymptotic value is reached in approximately TE-06 sec.
Apparent resistivity for co-axial and co-planar responses in a formation layer
comprising
multiple sub-layers
FIG. 47 shows an electromagnetic induction too180 in a formation layer I 10
comprising a sequence or package of alternating sets of sub-layers 112 and
114, set 112
having electromagnetic properties, notably conductivity, that is different
from set 114. The
tool axis is depicted in the plane of the sub-layers.
While each sub-layer in the laminate of thin layers may have isotropic
properties
such as isotropic conductivity, the combined effect of the sub-layers may be
that the
formation layer that consists of the sub-layers exhibits an anisotropic
electromagnetic
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induction. If each sub-layer 112, 114 in the formation layer 110 acts as an
individual
resistor, the macroscopic resistivity (inverse of conductivity) of the
formation layer in a
planar direction may be a resultant of all the layer-resistors in parallel
while the
macroscopic resistivity in a normal direction (i.e. perpendicular to the
layers) may be a
resultant of all the layer resistors in series.
In equation form:
1 0
(63) R, _A f R(z) = dz for the resistivity in the vertical, or principal
direction, and
(64) 6H =~f A 6(x) = dx for the conductivity in the horizontal, or in-plane,
direction
perpendicular to the principal direction. Of course, 6V can be found using 6V
= 1/RV, and
RH can be found using RH = 1/6H. Hence the in-plane resistivity is typically
lower than
the resistivity in the principal direction. These equations also hold for more
general cases
whereby the sub-layers are not of equal thickness and/or the sublayers are not
of equal
conductivity.
FIG. 48 shows the calculated apparent resistivity for the tool in the geometry
of
FIG. 47, whereby L = 1 m; the resistivity of sub-layers 112 is 10 S2m; the
resistivity of sub-
layers 114 is 1 S2m, and each sub-layer is 10 m of thickness. Line 115
corresponds to
apparent resistivity for co-axial measurement geometry while line 116
corresponds to
apparent resistivity for co-planar measurement geometry.
The apparent resistivity represented by lines 115 and 116 reflect the near-
layer
resistivity of 1 S2m at short times after the switching off of the
transmitter. After a time
span of approximately 2E-5 sec, the apparent resistivity starts to increase
due to the higher
resistivity of 10 S2m in the first adjacent sub-layers 112. So far, the
apparent resistivity
reflects what was set forth above for formations comprising two or three
isotropic
formation layers.
However, for later times the sub-layers are no longer individually resolved in
the
responses, in which case apparent resistivity is believed to reflect
contributions from the
sub-layer where the too180 is located, the adjacent layers and next adjacent
layers, and so
on. Effectively, the transient responses will show the macroscopic anisotropic
behavior. In
the example of Fig. 48, the collection of the isotropic sublayers that are not
individually
resolved in the transient responses are described by assuming an anisotropic
layer with an
anisotropic ratio of a2 = RH/RV = 1/((THRV) = 1/(0.55=5.5) = 0.33, which can
be found
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out using the late time apparent resistivities as set forth above for the
homogeneous
anisotropic formation. It is better to invert the responses assuming a
homogeneous
anisotropy than to try and determine the individual sub-layer structure.
The dotted lines 117 and 118 in Fig. 48, which correspond to the co-axial and
co-
planar apparent conductivities calculated for RH = 1.82 (i.e. 1/0.55) S2m and
RV = 5.5 S2m,
indeed match the drawn lines 115 and 116 well, at large t.
The combined, "macroscopic," anisotropic effect of a sub-layered anomaly, such
as
is shown in FIG. 49, may also be observed. Here, the anomaly A is formed of a
formation
layer having a thickness A comprising a thinly laminated sequence of a first
formation
material Al and a second formation material A2. FIG. 49 also indicates
coaxia160,
coplanar 62, and cross-component 64 measurement arrangements, wherein a
transmitter
coil T and a receiver coil R are spaced a distance L apart from each other.
The distance
between the transmitter coil T and the nearest interface 55 between the near
formation
layer and the anomaly A is indicated by D1.
Using the principles set forth above, the analysis taking into account
anisotropy
may be extended to multiple bedded formations, including those where only a
distant
formation layer displays macroscopic electromagnetic induction responses (such
as for
instance in FIG. 49) or where a local formation layer wherein the transmitter
and receiver
antennae are located, displays anisotropic behavior but whereby one or more
other,
isotropic or anisotropic layers are present at a distance.
Geosteering applications
As stated before in this specification, electromagnetic anisotropy may arise
intrinsically in certain types of formations, such as shales. A shale may cap
a reservoir of
mineral hydrocarbon fluids. It would thus be beneficial to precisely locate a
shale during
drilling of a well, and drill between for instance 10 m and 100 m below the
shale to enable
optimal production of the hydrocarbon fluids from the reservoir. This can be
done either by
traversing the shale or steering below the shale in a deviated well such as a
horizontal
section.
In other cases, the hydrocarbon containing reservoir may have materialized in
the
form of a stack of thin sands, which itself may exhibit anisotropic
electromagnetic
properties. It would be beneficial to identify the presence of such sands and
steer the
drilling bit into these sands.
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In each of these cases, geosteering may be accomplished by performing the
transient electromagnetic analysis while drilling and taking into account
formation
anisotropy. This may be implemented using the system as schematically depicted
in FIG.
1A.
More generally, geosteering decisions may be taken based on locating any type
of
electromagnetic anomaly using transient electromagnetic responses. Such
geosteering
applications allow to more accurately locate hydrocarbon fluid containing
reservoirs and to
more accurately drill into such reservoirs allowing to produce hydrocarbon
fluids from the
reservoirs with a minimum of water.
In order to produce the mineral hydrocarbon fluid from an earth formation, a
well
bore may be drilled with a method comprising the steps of:
suspending a drill string in the earth formation, the drill string comprising
at least a
drill bit and measurement sub comprising a transmitter antenna and a receiver
antenna;
drilling a well bore in the earth formation;
inducing an electromagnetic field in the earth formation employing the
transmitter
antenna;
detecting a transient electromagnetic response from the electromagnetic field,
employing the receiver antenna;
deriving a geosteering cue from the electromagnetic response.
Drilling of the well bore may then be continued in accordance with the
geosteering
cue until a reservoir containing the hydrocarbon fluid is reached.
Once the well bore extends into the reservoir containing the mineral
hydrocarbon
fluid, the well bore may be completed in any conventional way and the mineral
hydrocarbon fluid may be produced via the well bore.
Geosteering may be based on locating an electromagnetic anomaly in the earth
formation by analysing the transient response in accordance with the present
specification,
and taking a drilling decision based on the location relative to the
measurement sub. The
location of the anomaly may be expressed in terms of distance and/or direction
from the
measurement sub to the anomaly.
To facilitate executing the drilling decision, the drill string may comprise a
steerable drilling system 19, as shown in FIG. 1A. The drilling decision may
comprise
controlling the direction of drilling, e.g. by utilizing the steering system
19 if provided,
and/or establishing the remaining distance to be drilled.
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Accordingly, the geosteering cue may comprise information reflecting distance
between the target ahead of the bit and the bit, and/or direction from the bit
to target.
Distance and direction from the bit to the target may be calculated from the
distance and
direction from the tool to the bit, provided that the bit has a known location
relative to the
electromagnetic measurement tool.
Transient electromagnetic induction data may be correlated with the presence
of a
mineral hydrocarbon fluid containing reservoir, either directly by
establishing conductivity
values for the reservoir or indirectly by establishing quantitative
information on formation
layers that typically surround a mineral hydrocarbon fluid containing
reservoir.
In preferred embodiments, the transient electromagnetic induction data,
processed
in accordance with the above, is used to decide where to drill the well bore
and/or what is
its preferred path or trajectory. For instance, one may want to stay clear
from faults. Instead
of that, or in addition to that, it may be desirable to deviate from true
vertical drilling
and/or to steer into the reservoir at the correct depth.
The distance from the measurement sub to an anomaly in the formation may be
determined from the time in which one of apparent conductivity and apparent
resistivity
begins to deviate from the corresponding one of conductivity and resistivity
of formation in
which the measurement sub is located and/or determining time in which one of
apparent
dip and apparent azimuth and cross-component response starts to deviate from
zero. The
distance may also be determined from when one of apparent dip and apparent
azimuth
reaches an asymptotic value.
The electromagnetic anomaly may be located using at least one of time-
dependent
apparent conductivity, time dependent apparent resistivity, time-dependent dip
angle, and
time-dependent azimuth angle from the time dependence of the transient
response, in
accordance with the disclosure elsewhere hereinabove.
Any of the above mentioned time-dependencies can provide a useful geosteering
cue.
Fast Imaging Utilizing Apparent Conductivity and apparent angle
Apparent conductivity and apparent dip may also be used to create an "image"
or
representation of the formation features. This is accomplished by collecting
transient
apparent conductivity data at different positions within the borehole.
The apparent conductivity should be constant and equal to the formation
conductivity in a homogeneous formation. The deviation from a constant
conductivity
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value at time (t) suggests the presence of a conductivity anomaly in the
region specified by
time (t). The collected data may be used to create an image of the formation
relative to the
tool.
When the apparent resistivity plots (Rpp(z; t)) or apparent conductivity plots
((6pp(z; t)) at different tool positions are arranged together to form a plot
in both z- and t-
coordinates, the whole plot may be used as an image log to view the formation
geometry,
even if the layer resistivity may not be immediately accurately determined.
An example of such an image representation of the transient data as shown in
FIG.
50 for a L = 1 coaxial tool. The z coordinate references the tool depth along
the borehole.
The 6pp(z; t) plot shows the approaching bed boundary as the tool moves along
the
borehole.
FIG. 51 shows another example. The z-coordinate represents the tool depth
along
the borehole with the borehole intersecting the layer boundary in this case.
The 6app(z; t)
plot clearly helps to visualize the approaching and crossing the bed boundary
as the tool
moves along the borehole, for instance during drilling of the borehole.
Another
example is shown in FIG. 52 wherein a 3-layer model is used in conjunction
with a coaxial
tool having a 1 m spacing is in two differing positions in the formation. The
results are
plotted on FIG. 53A, where the apparent resistivity Rapp(t) is plotted at
various points as the
coaxial too180 approaches the resistive layer (see FIG. 53B).
FIG. 53A may be compared to FIG. 53B to discern the formation features.
Starting
in the 10 S2m layer 82, the drop in Rpp(t) is attributable to the 1 S2m layer
83 and the
subsequent increase in Rpp(t) is attributable to the 100 S2m layer 84. Curves
(91, 92, 93)
may readily be fitted to the inflection points to identify the responses to
the various beds,
effectively imaging the formation. Line 91 corresponds to the deflection
points caused by
the 1 S2m bed 83, line 92 to the salt 84, and line 93 to the deflection points
caused by 10
S2m bed 82. Moreover, the 1 S2m curve may be readily attributable to direct
signal pick up
between the transmitter and receiver when the tool is located in the 1 S2m
bed.
In still another example, the apparent dip Bpp(t) may be used to generate an
image
log. In FIG. 54A a coaxial tool is seen as approaching a highly resistive
formation at a dip
angle of approximately 30 degrees. The apparent dip response is shown in FIG.
54B.
Arrow 94 indicates a response to salt with the tool at Z = 100. As noted
previously, the
time at which the apparent dip response occurs is indicative of the distance
to the
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formation. When the responses for different distances are plotted together, a
curve may be
drawn indicative of the response as the tool approaches the bed, as shown in
FIG. 54B.
Summarising, the subterranean formation traversed by a wellbore may be imaged
using a tool comprising a transmitter for transmitting electromagnetic signals
through the
formation and a receiver for detecting response signals in a procedure
comprising steps
wherein
- the tool is brought to a first position inside the wellbore;
- the transmitter is energized to propagate an electromagnetic signal into the
formation;
- a response signal that has propagated through the formation is detected;
- a derived quantity is calculated for the formation based on the detected
response signal
for the formation;
- the derived quantity for the formation is plotted against time.
Then the tool is moved to at least one other position within the wellbore,
whereafter
the steps set out above are repeated. Optionally, this can be done again. Then
an image of
the formation within the subterranean formation is created based on the plots
of the derived
quantity.
Optionally tool is then again moved to at least one more other position within
the
wellbore and the whole procedure can be repeated again.
Creating the image of the formation features may include identifying one or
more
inflection points on each plotted derived quantity and fitting a curve to the
one or more
inflection points.
Thus an image of the formation may be created using apparent
conductivity/resistivity and apparent dip angle without the additional
processing required
for inversion and extraction of information. This information is capable of
providing
geosteering queues as well as the ability to profile subterranean formations.
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