Simultaneous inversion of induction data for dielectric permittivity and electric conductivity
A method of inverting induction logging data for evaluating the properties of underground formations surrounding a borehole, the data including induction voltage measurements obtained from a tool placed close to the formations of interest, the method includes: (a) defining a relationship relating the induction voltage to wave number, dielectric permittivity and conductivity; defining a cubic polynomial expansion of the relationship; and solving the cubic polynomial relationship using the voltage measurements to obtain values for conductivity that includes skin-effect correction, and apparent dielectric permittivity; and (b) using the obtained values for conductivity and apparent dielectric permittivity to derive a simulated value of induction voltage; determining the difference between the simulated value of the induction voltage and the measured induction voltage; and iteratively updating the values of conductivity and dielectric permittivity used for the derivation of the simulated value of induction voltage to minimize its difference with respect to the measured value.
1. A method of inverting induction logging data for evaluating the properties of underground formations surrounding a borehole, the data comprising induction voltage measurements obtained from a tool placed close to the formations of interest, the method comprising:
(a) operating a tool in a borehole to obtain induction voltage data;
(b) defining a relationship relating the induction voltage to wave number, dielectric permittivity and conductivity;
defining a cubic polynomial expansion of the relationship; and
solving the cubic polynomial relationship using the voltage measurements to obtain values for conductivity that includes skin-effect correction and apparent dielectric permittivity; and
(c) using the obtained values for conductivity and apparent dielectric permittivity to derive a simulated value of induction voltage;
determining the difference between the simulated value of the induction voltage and the measured induction voltage; and
using a higher-order polynomial expression to iteratively update the values of conductivity and dielectric permittivity used for the derivation of the simulated value of induction voltage to minimise its difference with respect to the measured value.
2. A method as claimed in claim 1 , wherein the relationship relating induction voltage to wave number, dielectric permittivity and conductivity is:
k
=
ω
c
μ
r
ɛ
r
+
ⅈ
σ
ωɛ
0
where k is wave number, ω is circular frequency, c is the speed of light in a vacuum, μ r is relative magnetic permeability, ∈ r is relative permittivity, ∈ 0 is the absolute dielectric permittivity of a vacuum and σ is conductivity.
3. A method as claimed in claim 1 , further comprising deriving roots of the cubic polynomial and using at least one of the roots to obtain the wave number.
4. A method as claimed in claim 3 , further comprising ignoring roots giving physically impossible values for parameters of interest.
5. A method as claimed in claim 1 , wherein the step of using a higher-order polynomial expression to iteratively update the values of conductivity and dielectric permittivity used for the derivation of the simulated value of induction voltage to minimise its difference with respect to the measured value, comprises minimising a squared difference between measured induction voltage U meas and a simulated reproduction U simul according to the relationship:
L
=
1
2
(
U
meas
-
U
simul
)
2
⇒
ⅆ
ⅆ
k
L
=
(
U
meas
-
U
simul
)
ⅆ
U
simul
ⅆ
k
=
0.
6. A method as claimed in claim 5 , wherein the higher order polynomial expression is a quadratic expression according to:
U
meas
=
U
simul
(
n
-
1
)
+
ⅆ
U
simul
(
n
-
1
)
ⅆ
k
(
k
(
n
)
-
k
(
n
-
1
)
)
+
1
2
ⅆ
2
U
simul
(
n
-
1
)
ⅆ
k
2
(
k
(
n
)
-
k
(
n
-
1
)
)
2
and the wave number k (n) is updated according to the relationship:
k
(
n
)
=
k
(
n
-
1
)
-
ⅆ
U
simul
(
n
-
1
)
/
ⅆ
k
ⅆ
2
U
simul
(
n
-
1
)
/
ⅆ
k
2
±
(
ⅆ
U
simul
(
n
-
1
)
/
ⅆ
k
ⅆ
2
U
simul
(
n
-
1
)
/
ⅆ
k
2
)
2
-
2
U
simul
(
n
-
1
)
-
U
meas
ⅆ
2
U
simul
(
n
-
1
)
/
ⅆ
k
2
.
7. A method as claimed in claim 6 , further comprising deriving roots of the cubic polynomial and ignoring roots giving physically impossible values for parameters of interest.
8. A method as claimed in claim 5 , wherein the higher order polynomial expression is a cubic expression according to:
U
meas
=
U
simul
(
n
-
1
)
+
ⅆ
U
simul
(
n
-
1
)
ⅆ
k
(
k
(
n
)
-
k
(
n
-
1
)
)
+
1
2
ⅆ
2
U
simul
(
n
-
1
)
ⅆ
k
2
(
k
(
n
)
-
k
(
n
-
1
)
)
2
+
1
6
ⅆ
3
U
simul
(
n
-
1
)
ⅆ
k
3
(
k
(
n
)
-
k
(
n
-
1
)
)
3
and the wave number k (n) is updated according to the relationship:
(
k
(
n
)
-
k
(
n
-
1
)
)
3
+
3
ⅆ
2
U
simul
(
n
-
1
)
/
ⅆ
k
2
ⅆ
3
U
simul
(
n
-
1
)
/
ⅆ
k
3
(
k
(
n
)
-
k
(
n
-
1
)
)
2
+
6
ⅆ
U
simul
(
n
-
1
)
/
ⅆ
k
ⅆ
3
U
simul
(
n
-
1
)
/
ⅆ
k
3
(
k
(
n
)
-
k
(
n
-
1
)
)
+
6
U
simul
(
n
-
1
)
-
U
meas
ⅆ
3
U
simul
(
n
-
1
)
/
ⅆ
k
3
=
0.
9. A method as claimed in claim 8 , further comprising deriving roots of the cubic polynomial and ignoring roots giving physically impossible values for parameters of interest.
10. A method as claimed in claim 1 , wherein the induction voltage measurements comprise in-phase and quadrature measurements of substantially the same accuracy and resolution.
11. A method as claimed in claim 1 , wherein the induction logging data is obtained from a three-coil tool comprising a transmitter coil with magnetic dipole moment M T , a main receiver coil with magnetic dipole moment M 1 at distance r 1 from the transmitter coil, and a bucking coil with a magnetic moment M 2 at distance r 2 from the transmitter coil, the dipole moments being aligned substantially parallel to the axis of the tool, the cubic polynomial expansion comprising:
U
l
=
-
ⅈωμ
M
T
2
π
(
(
M
1
2
r
1
-
M
2
2
r
2
)
k
2
+
ⅈ
(
M
1
3
-
M
2
3
)
k
3
)
wherein U l is the induction voltage.
12. A method as claimed in claim 11 , wherein the simulated value of induction voltage is derived according to the relationship
U
l
=
-
ⅈωμ
M
T
2
π
(
M
1
ⅇ
ⅈ
kr
1
r
1
3
(
1
-
ⅈ
kr
1
)
-
M
2
ⅇ
ⅈ
kr
2
r
2
3
(
1
-
ⅈ
kr
2
)
)
using a sensitivity function derived according to the relationship:
ⅆ
U
l
ⅆ
k
=
-
ⅈωμ
M
T
2
π
(
M
1
ⅇ
ⅈ
kr
1
r
1
-
M
2
ⅇ
ⅈ
kr
2
r
2
)
k
where e ikr is the full electromagnetic wave.
13. A method as claimed in claim 1 , wherein the induction logging data is obtained from a three-coil tool comprising a transmitter coil with magnetic dipole moment M T , a main receiver coil with magnetic dipole moment M 1 at distance r 1 from the transmitter coil, and a bucking coil with a magnetic moment M 2 at distance r 2 from the transmitter coil, the dipole moments being aligned substantially transverse to the axis of the tool, the cubic polynomial expansion comprising:
U
t
=
-
ⅈωμ
M
T
4
π
(
(
M
1
2
r
1
-
M
2
2
r
2
)
k
2
+
2
ⅈ
(
M
1
3
-
M
2
3
)
k
3
)
wherein U t is the induction voltage.
14. A method as claimed in claim 13 , wherein the simulated value of induction voltage is derived according to the relationship
U
t
=
-
ⅈωμ
M
T
4
π
(
M
1
ⅇ
ⅈ
kr
1
r
1
3
(
1
-
ⅈ
kr
1
-
k
2
r
1
2
)
-
M
2
ⅇ
ⅈ
kr
2
r
2
3
(
1
-
ⅈ
kr
2
-
k
2
r
2
2
)
)
using a sensitivity function derived according to the relationship:
ⅆ
U
t
ⅆ
k
=
ⅈωμ
M
T
4
π
(
M
1
ⅇ
ⅈ
kr
1
r
1
(
1
+
ⅈ
kr
1
)
-
M
2
ⅇ
ⅈ
kr
2
r
2
(
1
+
ⅈ
kr
2
)
)
k
where e ikr is the full electromagnetic wave.
15. A method of inverting induction logging data for evaluating the properties of underground formations surrounding a borehole, the data comprising induction voltage measurements obtained from a tool placed close to the formations of interest, the method comprising:
(a) operating a tool in a borehole to obtain induction voltage data;
(b) defining a relationship relating the induction voltage to wave number, dielectric permittivity and conductivity;
defining a n>2 polynomial expansion of the relationship, wherein the n>2 polynomial expansion comprises a cubic polynomial expansion, a higher-order polynomial expansion, or combinations thereof; and
solving the n>2 polynomial relationship using the voltage measurements to obtain values for conductivity that includes skin-effect correction and apparent dielectric permittivity; and
(c) using the obtained values for conductivity and apparent dielectric permittivity to derive a simulated value of induction voltage;
determining the difference between the simulated value of the induction voltage and the measured induction voltage; and
using a n+1 polynomial expression to iteratively update the values of conductivity and dielectric permittivity used for the derivation of the simulated value of induction voltage to minimise its difference with respect to the measured value.
16. A method as claimed in claim 15 , wherein the relationship relating induction voltage to wave number, dielectric permittivity and conductivity is:
k
=
ω
c
μ
r
ɛ
r
+
ⅈ
σ
ωɛ
0
where k is wave number, ω is circular frequency, c is the speed of light in a vacuum, μ r is relative magnetic permeability, ∈ r is relative permittivity, ∈ 0 is the absolute dielectric permittivity of a vacuum and σ is conductivity.
17. A method as claimed in claim 15 , further comprising deriving roots of the n>2 polynomial and using at least one of the roots to obtain the wave number.
18. A method as claimed in claim 17 , further comprising ignoring roots giving physically impossible values for parameters of interest.
19. A method as claimed in claim 15 , wherein the induction voltage measurements comprise in-phase and quadrature measurements of substantially the same accuracy and resolution.
20. A method as claimed in claim 15 , wherein the induction logging data is obtained from a three-coil tool comprising a transmitter coil with magnetic dipole moment M T , a main receiver coil with magnetic dipole moment M 1 at distance r 1 from the transmitter coil, and a bucking coil with a magnetic moment M 2 at distance r 2 from the transmitter coil, the dipole moments being aligned substantially parallel to the axis of the tool, the cubic polynomial expansion comprising:
U
l
=
-
ⅈωμ
M
T
2
π
(
(
M
1
2
r
1
-
M
2
2
r
2
)
k
2
+
ⅈ
(
M
1
3
-
M
2
3
)
k
3
)
wherein U l is the induction voltage.