Cross-coupling based determination of anisotropic formation properties
Tools, systems, and methods are disclosed for multi-component induction logging with iterative analytical conversion of tool measurements to formation parameters. At least some system embodiments include a logging tool and at least one processor. The logging tool provides transmitter-receiver coupling measurements that include at least diagonal coupling measurements (Hzz, Hxx, and/or Hyy) and cross-coupling measurements (Hxy, Hxz, and Hyz). The processor employs an iterative analytical conversion of the cross-coupling measurements into formation resistive anisotropy and dip information. The processor may further provide one or more logs of the resistive anisotropy and/or dip information.
1. A method for determining conductivity in a formation, wherein the method comprises:
gathering, by a borehole logging tool, multi-component transmitter-receiver coupling measurements using a transmitter and at least one receiver spaced from the transmitter, wherein the transmitter comprises a triad of mutually orthogonal transmitters and the receiver comprises a triad of mutually orthogonal receivers with a triad of bucking coils;
receiving multi-component transmitter-receiver coupling measurements including diagonal coupling measurements and cross-coupling measurements gathered by the borehole logging tool;
obtaining, by a processor, from the cross-coupling measurements and not from the diagonal coupling measurements: a strike angle, an initial dip angle, an initial anisotropy factor, and an initial horizontal conductivity;
determining, by the processor, from the cross-coupling measurements, the strike angle, the latest dip angle, and the latest anisotropy factor, and the latest conductivity: an iterative anisotropy factor, an iterative horizontal conductivity, and an iterative dip angle; and
displaying a log based at least in part on at least one of said iterative values.
2. The method of claim 1 , wherein the initial horizontal conductivity σ 1 corresponds to
σ
1
=
K
1
[
(
H
xz
)
X
-
(
H
xz
)
R
]
(
1
-
A
1
)
tan
α
1
cos
β
,
wherein (H zx ) X and (H zx ) R are the imaginary and real parts, respectively, of the H zx cross-coupling measurement, K 1 is a tool constant, A 1 is the initial anisotropy factor corresponding to
A
1
=
1
1
+
(
H
xy
)
X
-
(
H
xy
)
R
(
H
xz
)
X
-
(
H
xz
)
R
*
tan
α
1
sin
β
,
wherein (H xy ) X and (H xy ) R are the imaginary and real parts, respectively, of the H xy cross-coupling measurement, α 1 is the initial dip angle corresponding to
α
1
=
arctan
[
(
(
H
xy
)
R
(
H
xz
)
R
)
2
+
(
(
H
xy
)
R
(
H
yz
)
R
)
2
]
,
wherein (H zy ) R is the real parts of the H zy cross-coupling measurement, and wherein β is the strike angle corresponding to
β
=
arctan
(
H
yz
t
H
xz
t
)
.
3. The method of claim 1 , wherein the iterative horizontal conductivity σ i+1 corresponds to
σ
i
+
1
=
K
1
[
(
H
xz
)
X
-
(
H
xz
)
R
]
(
1
-
A
i
+
1
)
tan
α
i
+
1
cos
β
,
wherein (H zx ) X and (H zx ) R are the imaginary and real parts, respectively, of the H zx cross-coupling measurement, K 1 is a tool constant, A i+1 corresponds to
A
i
+
1
=
1
1
+
(
H
xy
)
X
-
(
H
xy
)
R
-
Γ
xy
(
H
xz
)
X
-
(
H
xz
)
R
-
Γ
xz
*
tan
α
i
sin
β
,
wherein (H xy ) X and (H xy ) R are the imaginary and real parts, respectively, of the H xy cross-coupling measurement, Γ xy and Γ zx are iterative corrections, α i+1 corresponds to
α
i
+
1
=
arctan
[
(
(
H
xy
)
R
-
Λ
xy
(
H
xz
)
R
-
Λ
xz
)
2
+
(
(
H
xy
)
R
-
Λ
xy
(
H
yz
)
R
-
Λ
yz
)
2
]
,
wherein (H zy ) R is the real parts of the H zy cross-coupling measurement, and Λ xy , Λ zx , and Λ zy are iterative correction terms, and wherein β is the strike angle corresponding to
β
=
arctan
(
H
yz
t
H
xz
t
)
.
4. The method of claim 3 , wherein the iterative correction Γ zx corresponds to
Γ
xz
=
cos
β
sin
2
α
i
sin
2
α
i
[
-
2
3
(
1
-
A
i
3
)
{
K
2
(
L
M
δ
i
)
4
+
K
3
(
L
B
δ
i
)
4
}
+
1
3
(
1
-
A
i
4
)
{
K
2
(
L
M
δ
i
)
5
+
K
3
(
L
B
δ
i
)
5
}
]
and the iterative correction Γ xy corresponds to
Γ
xy
=
sin
2
β
sin
2
α
i
[
-
T
1
{
K
2
(
L
M
δ
i
)
4
+
K
3
(
L
B
δ
i
)
4
}
-
2
T
2
{
K
2
(
L
M
δ
i
)
5
+
K
3
(
L
B
δ
i
)
5
}
]
,
with
T
1
=
2
3
(
cos
2
α
i
+
1
)
(
A
i
3
-
1
)
-
2
A
i
(
A
i
2
-
cos
2
α
i
)
+
2
sin
2
α
i
T
2
=
-
1
6
(
cos
2
α
i
+
1
)
(
A
i
4
-
1
)
+
2
3
A
i
2
(
A
i
2
-
cos
2
α
i
)
-
2
3
sin
2
α
i
wherein β is the strike angle, α i is the latest dip angle, A i is the latest anistropy factor, K 2 and K 3 are tool constants, L M and L B are distances between the transmitter and the receiver's main coils and bucking coils, respectively, and δ i =√{square root over (2/ωμσ i )} is the latest skin depth, with σ i being the latest conductivity.
5. The method of claim 3 , wherein the iterative correction term Λ zx corresponds to
Λ
zx
=
cos
βsin
2
α
i
sin
2
α
i
[
2
3
(
1
-
A
i
+
1
3
)
(
K
2
(
L
M
δ
i
+
1
)
4
+
K
3
(
L
B
δ
i
+
1
)
4
)
-
1
6
(
1
-
A
i
+
1
4
)
(
K
2
(
L
M
δ
i
+
1
)
5
+
K
3
(
L
B
δ
i
+
1
)
5
)
]
the iterative correction term Λ zy corresponds to
Λ
zy
=
sin
βsin
2
α
i
sin
2
α
i
[
2
3
(
1
-
A
i
+
1
3
)
(
K
2
(
L
M
δ
i
+
1
)
4
+
K
3
(
L
B
δ
i
+
1
)
4
)
-
1
6
(
1
-
A
i
+
1
4
)
(
K
2
(
L
M
δ
i
+
1
)
5
+
K
3
(
L
B
δ
i
+
1
)
5
)
]
and the iterative correction term Λ xy corresponds to
Λ
xy
=
sin
2
β
sin
2
α
i
[
T
1
(
K
2
(
L
M
δ
i
+
1
)
4
+
K
3
(
L
B
δ
i
+
1
)
4
)
+
T
2
(
K
2
(
L
M
δ
i
+
1
)
5
+
K
3
(
L
B
δ
i
+
1
)
5
)
]
,
with
T
1
=
2
3
(
cos
2
α
i
+
1
)
(
A
i
+
1
3
-
1
)
-
2
A
i
+
1
(
A
i
+
1
2
-
cos
2
α
i
)
+
2
sin
2
α
i
T
2
=
-
1
6
(
cos
2
α
i
+
1
)
(
A
i
+
1
4
-
1
)
+
2
3
A
i
+
1
2
(
A
i
+
1
2
-
cos
2
α
i
)
-
2
3
sin
2
α
i
wherein is the strike angle, α i is the latest dip angle, A i+1 is the latest anistropy factor, K 2 and K 3 are tool constants, L M and L B are distances between the transmitter and the receiver's main coils and bucking coils, respectively, and δ i =√{square root over (2/ωμσ i )} is the latest skin depth, with σ i being the latest conductively.
6. A logging method that comprises:
conveying a logging tool along a borehole through a formation, wherein the logging tool comprises:
a transmitter, wherein the transmitter further comprises a triad of mutually orthogonal transmitters; and
at least one receiver spaced from the transmitter, wherein the at least one receiver comprises a triad of mutually orthogonal receivers with a triad of bucking;
inducing an electromagnetic field into a formation;
measuring an in-phase and quadrature phase voltage signal induced in each of the at least one receiver coils by the transmitter;
producing a multicomponent measurement from the recording the voltage with the at least one receiver;
producing a diagonal coupling measurement from the multicomponent measurement;
producing a cross-coupling measurement form the multicomponent measurement;
obtaining, by a processor, from the cross-coupling measurements and not from the diagonal coupling measurements: a strike angle, an initial dip angle, an initial anisotropy factor, and an initial conductivity;
determining, by the processor, from the cross-coupling measurements, the strike angle, the latest dip angle, and the latest anisotropy factor, and the latest conductivity: an iterative anisotropy factor, an iterative conductivity, and an iterative dip angle; and
displaying a log based at least in part on at least one of said iterative values.
7. The method of claim 6 , further comprising:
repeating said determining of said iterative anisotropy factor, iterative conductivity, and iterative dip angle.
8. The method of claim 7 , wherein said repeating is performed a number of times that minimizes an overall residual error.