Extended 1D inversion of electromagnetic measurements for formation evaluation
A method to determine at least one formation property of a subterranean formation includes providing a downhole electromagnetic logging tool having at least one transmitter array and one receiver array to acquire measurements in the formation. The method further includes performing a first inversion in response to the measurements, assuming dip or dip azimuth is constant in an inversion zone within the formation to obtain an inverted formation model. The method includes determining an n-th order partial derivative matrix of at least one of dip or dip azimuth, wherein n is greater than or equal to 1. The method includes performing a second inversion using the determined n-th order partial derivative matrix, wherein at least one of dip and dip azimuth are allowed to vary in the inversion zone, to obtain an updated formation model. The method includes determining the formation property using the updated formation model.
1. A method to determine at least one formation property of a subterranean formation, comprising:
providing a downhole electromagnetic logging tool having at least one transmitter array and one receiver array;
acquiring measurements in the formation using the transmitter and receiver arrays of the downhole electromagnetic logging tool;
performing a first inversion in response to the measurements, wherein at least one of dip or dip azimuth are assumed constant in an inversion zone within the formation to obtain an inverted formation model that comprises at least one of horizontal resistivity (Rh), vertical resistivity (Rv), dip, and dip azimuth;
determining an n-th order partial derivative matrix of at least one of dip or dip azimuth, wherein n is greater than or equal to 1;
performing a second inversion using the determined n-th order partial derivative matrix, wherein at least one of dip and dip azimuth are allowed to vary in the inversion zone, to obtain an updated formation model; and
determining the at least one formation property of the formation using the updated formation model.
2. The method of claim 1 , wherein n=1 and the partial derivative matrix comprises a Jacobian matrix of at least one of dip or dip azimuth.
3. The method of claim 1 , wherein determining the n-th order partial derivative matrix comprises using a mathematical expansion.
4. The method of claim 3 , wherein the mathematical expansion comprises using an approximated expansion that ignores contribution of at least one of dip or dip azimuth from beds in the inverted formation model other than a host bed in which a sensitivity center of the measurements is located.
5. The method of claim 3 , wherein the mathematical expansion comprises using a complete expansion that takes into account contributions of at least one of dip or dip azimuth from each bed in the inverted formation model.
6. The method of claim 3 , wherein the mathematical expansion comprises a Taylor series expansion.
7. The method of claim 5 , wherein the mathematical expansion is performed according to:
σ
a
(
α
,
β
)
≈
σ
a
(
α
0
,
β
0
)
+
∑
i
=
1
N
∂
σ
a
(
α
i
,
0
,
β
i
,
0
)
∂
α
i
(
α
i
-
α
0
)
+
∑
i
=
1
N
∂
σ
a
(
α
i
,
0
,
β
i
,
0
)
∂
β
i
(
β
i
-
β
0
)
wherein α 0 =(α k,0 ), β 0 =(β k,0 ), where α k,0 =α 0 , and β k,0 =β 0 , wherein α 0 and β 0 represents the dip and dip azimuth, respectively, from the inverted formation model, and wherein the n-th order partial derivative matrix is determined as a first-order partial derivative matrix that includes first-order partial derivatives for dip and dip azimuth determined using:
∂
H
n
T
n
R
∂
α
i
=
I
m
l
(
σ
v
,
i
-
σ
h
,
i
)
∫
V
i
d
r
_
′
(
E
x
,
n
R
E
z
,
n
T
+
E
x
,
n
T
E
z
,
n
R
)
∂
H
n
T
n
R
∂
β
i
=
I
m
l
(
σ
v
,
i
-
σ
h
,
i
)
sin
θ
∫
V
i
d
r
_
′
(
E
y
,
n
R
E
z
,
n
T
+
E
y
,
n
T
E
z
,
n
R
)
wherein H n R n T is a magnetic field at a receiver of the receiver array oriented in the nR direction in response to a transmitter of the transmitter array oriented in the nr direction, I m l is a dipole moment of a source corresponding to the transmitter, E n T =(E x,n T ,E y,n T ,E z,n T ) t is an electric field due to the source with a unit dipole moment corresponding to the transmitter, E n R =(E x,n R ,E y,n R ,E z,n R ) t is an electric field due to a source with a unit dipole moment corresponding to the receiver, σ h,i and σ v,i represent horizontal conductivity and vertical conductivity of bed i, respectively, θ is a bedding dip in accordance with a tool coordinate system, V i designates the support of bed i, and α i and β i represent dip and dip azimuth, respectively of bed i.
8. The method of claim 1 , comprising using at least one of dip or dip azimuth from the updated formation model to find variation in at least one of dip or azimuth with respect to measured depth or true vertical depth to determine complexity of the formation.
9. The method of claim 1 , wherein n=2 and the partial derivative matrix comprises a Hessian matrix of at least one of dip or dip azimuth.
10. A system comprising:
an electromagnetic logging tool having at least one transmitter array and one receiver array, the electromagnetic logging tool being configured to acquire measurements in a subterranean formation using the transmitter and receiver arrays when disposed in the formation; a processing device configured to:
perform a first inversion in response to the measurements wherein at least one of dip or dip azimuth are assumed constant in an inversion zone within the formation to obtain an inverted formation model that comprises at least one of horizontal resistivity (Rh), vertical resistivity (Rv), dip, and dip azimuth;
determine an n-th order partial derivative matrix of at least one of dip or dip azimuth, wherein n is greater than or equal to 1;
perform a second inversion using the determined n-th order partial derivative matrix, wherein at least one of dip and dip azimuth are allowed to vary in the inversion zone, to obtain an updated formation model; and
determine at least one formation property of the formation using the updated formation model.
11. The system of claim 10 , wherein the electromagnetic logging tool is conveyed using at least one of drill pipe, wireline, slickline, or coiled tubing.
12. The system of claim 10 , wherein the electromagnetic logging tool comprises a wireline or an logging-while-drilling tool.
13. The system of claim 10 , wherein n=1 and the partial derivative matrix comprises a Jacobian matrix of at least one of dip or dip azimuth.
14. The system of claim 10 , wherein the processing device determines the n-th order partial derivative matrix comprises using a mathematical expansion.
15. The system of claim 14 , wherein the mathematical expansion comprises using an approximated expansion that ignores contribution of at least one of dip or dip azimuth from beds in the inverted formation model other than a host bed in which a sensitivity center of the measurements is located.
16. The system of claim 14 , wherein the mathematical expansion comprises using a complete expansion that takes into account contributions of at least one of dip or dip azimuth from each bed in the inverted formation model.
17. The system of claim 14 , wherein the mathematical expansion comprises a Taylor series expansion.
18. A non-transitory computer-readable medium having computer executable instructions for causing a computer to:
in response to electromagnetic measurement obtained by an electromagnetic logging tool disposed in a formation, perform a first inversion wherein at least one of dip or dip azimuth are assumed constant in an inversion zone within the formation to obtain an inverted formation model that comprises at least one of horizontal resistivity (Rh), vertical resistivity (Rv), dip, and dip azimuth;
determine an n-th order partial derivative matrix of at least one of dip or dip azimuth, wherein n is greater than or equal to 1;
perform a second inversion using the determined n-th order partial derivative matrix, wherein at least one of dip and dip azimuth are allowed to vary in the inversion zone, to obtain an updated formation model; and
determine at least one formation property of the formation using the updated formation model.
19. The non-transitory computer-readable medium of claim 18 , wherein the determination of the n-th order partial derivative matrix comprises using a mathematical expansion.
20. The non-transitory computer-readable medium of claim 19 , wherein the mathematical expansion comprises using an approximated expansion that ignores contribution of at least one of dip or dip azimuth from beds in the inverted formation model other than a host bed in which a sensitivity center of the measurements is located.
21. The non-transitory computer-readable medium of claim 19 , wherein the mathematical expansion comprises using a complete expansion that takes into account contributions of at least one of dip or dip azimuth from each bed in the inverted formation model.