Method of directional resistivity logging
A method for estimating at least one formation parameter from a directional resistivity measurement includes computing a plurality of hypothetical directional resistivity values at a corresponding plurality of formation parameter values. The computation makes use of a forward model having at least one analytical expression that relates a directional resistivity measurement to the formation parameter. The analytical expression includes at least one image source term. Comparison of computed directional resistivity values with measured direct resistivity values enables a value of at least one formation parameter to be selected. The method may be implemented on a downhole processor.
1. A method for estimating a value of at least a first formation parameter from a directional resistivity measurement, the method comprising:
(a) acquiring a forward model of a subterranean formation, the forward model including at least one analytical expression relating a directional resistivity measurement to at least the first formation parameter, wherein the analytical expression includes at least one image source term;
(b) causing a processor to compute a plurality of hypothetical directional resistivity measurements for a corresponding plurality of values of the first formation parameter;
(c) acquiring at least one directional resistivity measurement;
(d) selecting a value of at least the first formation parameter for which the hypothetical directional resistivity measurement computed in (b) substantially matches the directional resistivity measurement acquired in (c).
2. The method of claim 1 , wherein (b) comprises causing a downhole processor to compute a plurality of hypothetical directional resistivity measurements for a corresponding plurality of values of the first formation parameter.
3. The method of claim 1 , wherein the forward model acquired in (a) comprises a two-layer formation model in which a directional resistivity tool is deployed in a first formation layer spaced apart from a boundary between the first formation layer and a second formation layer.
4. The method of claim 3 , wherein the directional resistivity tool is substantially parallel with the boundary between the first and second formation layers.
5. The method of claim 4 , wherein the at least one analytical expression comprises an expression relating at least one cross-coupling component to at least the first formation parameter, the expression being selected from at least one of the following:
H
ZX
=
M
Z
4
π
ⅇ
-
j
k
n
r
r
5
(
k
n
2
r
2
-
3
j
k
n
r
-
3
)
(
2
d
+
2
h
)
a
︸
from
image
source
H
XZ
=
-
M
X
4
π
ⅇ
-
j
k
n
r
r
5
(
k
n
2
r
2
-
3
j
k
n
r
-
3
)
(
2
d
+
2
h
)
a
︸
from
image
source
wherein H ZX and H XZ represent the cross-coupling components, M Z and M X represent the dipole moments of z-mode and x-mode transmitters, a represents an axial distance between a transmitter and a receiver, r represents a distance between an image source and the receiver, d represents a complex valued distance between the boundary and a theoretical image plane, h represents a distance between directional resistivity tool and the boundary, and k n is the wave number of the first formation layer.
6. The method of claim 4 , wherein the at least one analytical expression comprises an expression relating at least one coupling component to at least the first formation parameter, the expression being selected from at least one of the following:
H
ZZ
=
M
Z
4
π
{
ⅇ
-
j
k
n
r
r
5
(
k
n
2
r
2
-
3
j
k
n
r
-
3
)
(
2
d
+
2
h
)
2
+
2
ⅇ
-
j
k
n
r
r
3
(
j
k
n
r
+
1
)
}
︸
from
image
source
+
M
Z
4
π
2
ⅇ
-
j
k
n
a
a
3
(
j
k
n
a
+
1
)
︸
from
real
source
H
XX
=
M
X
4
π
{
ⅇ
-
j
k
n
r
r
5
(
k
n
2
r
2
-
3
j
k
n
r
-
3
)
(
2
d
+
2
h
)
2
-
ⅇ
-
j
k
n
r
r
3
(
k
n
2
r
2
-
j
k
n
r
-
1
)
}
︸
from
image
source
+
M
X
4
π
ⅇ
-
j
k
n
a
a
3
(
k
n
2
a
2
-
j
k
n
a
-
1
)
︸
from
real
source
wherein H ZZ and H XX the coupling components, M Z and M X represent the dipole moments of z-mode and x-mode transmitters, a represents an axial distance between a transmitter and a receiver, r represents a distance between an image source and the receiver, d represents a complex valued distance between the boundary and a theoretical image plane, h represents a distance between directional resistivity tool and the boundary, and k n is the wave number of the first formation layer.
7. The method of claim 4 , wherein the at least one analytical expression comprises an expression relating at least one coupling component to at least the first formation parameter, the expression being selected from at least one of the following:
H
ZZ
=
M
Z
4
π
{
ⅇ
-
j
k
n
r
r
5
(
k
n
2
r
2
-
3
j
k
n
r
-
3
)
(
2
d
+
2
h
)
2
-
ⅇ
-
j
k
n
r
r
3
(
k
n
2
r
2
+
j
k
n
r
+
1
)
cos
2
θ
+
2
ⅇ
-
j
k
n
r
r
3
(
j
k
n
r
+
1
)
}
︸
from
image
source
+
M
Z
4
π
2
ⅇ
-
j
k
n
a
a
3
(
j
k
n
a
+
1
)
︸
from
real
source
H
XX
=
M
X
4
π
{
ⅇ
-
j
k
n
r
r
5
(
k
n
2
r
2
-
3
j
k
n
r
-
3
)
(
2
d
+
2
h
)
2
-
ⅇ
-
j
k
n
r
r
3
(
k
n
2
r
2
+
j
k
n
r
+
1
)
sin
2
θ
+
2
ⅇ
-
j
k
n
r
r
3
(
j
k
n
r
+
1
)
}
︸
from
image
source
+
M
X
4
π
ⅇ
-
j
k
n
a
a
3
(
k
n
2
a
2
-
j
k
n
a
-
1
)
︸
from
real
source
wherein H ZZ and H XX the coupling components, M Z and M X represent the dipole moments of z-mode and x-mode transmitters, a represents an axial distance between a transmitter and a receiver, r represents a distance between an image source and the receiver, d represents a complex valued distance between the boundary and a theoretical image plane, h represents a distance between directional resistivity tool and the boundary, k n is the wave number of the first formation layer, and θ represents a dip angle between the directional resistivity tool and the boundary.
8. The method of claim 4 , wherein the at least one analytical expression comprises an expression relating at least one coupling component to at least the first formation parameter, the expression being selected from at least one of the following:
H
ZZ
=
M
Z
4
π
{
ⅇ
-
j
k
n
r
1
r
1
5
(
k
n
2
r
1
2
-
3
j
k
n
r
1
-
3
)
(
2
d
1
+
2
h
1
)
2
+
2
ⅇ
-
j
k
n
r
1
r
1
3
(
j
k
n
r
1
+
1
)
}
+
︸
from
image
source
1
M
Z
4
π
{
ⅇ
-
j
k
n
r
2
r
2
5
(
k
n
2
r
2
2
-
3
j
k
n
r
2
-
3
)
(
2
d
2
+
2
h
2
)
2
+
2
ⅇ
-
j
k
n
r
2
r
2
3
(
j
k
n
r
2
+
1
)
}
︸
from
image
source
2
+
M
Z
4
π
2
ⅇ
-
j
k
n
a
a
3
(
j
k
n
a
+
1
)
︸
from
real
source
H
XX
=
M
X
4
π
{
ⅇ
-
j
k
n
r
1
r
1
5
(
k
n
2
r
1
2
-
3
j
k
n
r
1
-
3
)
(
2
d
1
+
2
h
1
)
2
-
ⅇ
-
j
k
n
r
1
r
1
3
(
k
n
2
r
1
2
-
j
k
n
r
1
-
1
)
}
+
︸
from
image
source
1
M
X
4
π
{
ⅇ
-
j
k
n
r
2
r
2
5
(
k
n
2
r
2
2
-
3
j
k
n
r
2
-
3
)
(
2
d
2
+
2
h
2
)
2
-
ⅇ
-
j
k
n
r
2
r
2
3
(
k
2
2
r
2
2
-
j
k
n
r
2
-
1
)
}
︸
from
image
source
2
+
M
X
4
π
ⅇ
-
j
k
n
a
a
3
(
k
n
2
a
2
-
j
k
n
a
-
1
)
︸
from
real
source
wherein H ZZ and H XX the coupling components, M Z and M X represent the dipole moments of z-mode and x-mode transmitters, a represents an axial distance between a transmitter and a receiver, r 1 and r 2 represent distances between first and second image sources and the receiver, d 1 and d 2 represent first and second complex valued distances between the first boundary and a first theoretical image plane and the second boundary and a second theoretical image plane, h 1 and h 2 represent distances between the directional resistivity tool and the first and second boundaries, and k n is the wave number of the first formation layer.
9. The method of claim 3 , wherein the at least one analytical expression comprises an expression relating at least one cross-coupling component to at least the first formation parameter, the expression being selected from at least one of the following:
H
ZX
=
M
Z
4
π
{
ⅇ
-
j
k
n
r
r
5
(
k
n
2
r
2
-
3
j
k
n
r
-
3
)
(
2
d
+
2
h
)
a
sin
θ
+
ⅇ
-
j
k
n
r
r
3
(
k
n
2
r
2
+
j
k
n
r
+
1
)
sin
θ
cos
θ
︸
from
image
source
H
XZ
=
M
X
4
π
{
-
ⅇ
-
j
k
n
r
r
5
(
k
n
2
r
2
-
3
j
k
n
r
-
3
)
(
2
d
+
2
h
)
a
sin
θ
+
ⅇ
-
j
k
n
r
r
3
(
k
n
2
r
2
+
j
k
n
r
+
1
)
sin
θcos
θ
︸
from
image
source
wherein H ZX and H XZ represent the cross-coupling components, M Z and M X represent the dipole moments of z-mode and x-mode transmitters, a represents an axial distance between a transmitter and a receiver, r represents a distance between an image source and the receiver, d represents a complex valued distance between the boundary and a theoretical image plane, h represents a distance between directional resistivity tool and the boundary, k n is the wave number of the first formation layer, and θ represents a dip angle between the directional resistivity tool and the boundary.
10. The method of claim 1 , wherein the forward model acquired in (a) comprises a three-layer model in which a directional resistivity tool is deployed in a first formation layer between first and second boundaries, the first boundary between the first formation layer and a second formation layer and the second boundary between the first formation layer and a third formation layer.
11. The method of claim 10 , wherein the first boundary and the second boundary are substantially parallel with one another.
12. The method of claim 11 , wherein the directional resistivity tool is substantially parallel with the first and second boundaries.
13. The method of claim 12 , wherein the at least one analytical expression comprises an expression relating at least one cross-coupling component to at least the first formation parameter, the expression being selected from at least one of the following:
H
ZX
=
M
Z
4
π
{
ⅇ
-
j
k
n
r
1
r
1
5
(
k
n
2
r
1
2
-
3
j
k
n
r
1
-
3
)
(
2
d
1
+
2
h
1
)
a
︸
from
image
source
1
+
M
Z
4
π
{
-
ⅇ
-
j
k
n
r
2
r
2
5
(
k
n
2
r
2
2
-
3
j
k
n
r
2
-
3
)
(
2
d
2
+
2
h
2
)
a
︸
from
image
source
2
H
XZ
=
M
X
4
π
{
-
ⅇ
-
j
k
n
r
1
r
1
5
(
k
n
2
r
1
2
-
3
j
k
n
r
1
-
3
)
(
2
d
1
+
2
h
1
)
a
︸
from
image
source
1
+
M
X
4
π
{
ⅇ
-
j
k
n
r
2
r
2
5
(
k
n
2
r
2
2
-
3
j
k
n
r
2
-
3
)
(
2
d
2
+
2
h
2
)
a
︸
from
image
source
2
wherein H ZX and H XZ represent the cross-coupling components, M Z and M X represent the dipole moments of z-mode and x-mode transmitters, a represents an axial distance between a transmitter and a receiver, r 1 and r 2 represent distances between first and second image sources and the receiver, d 1 and d 2 represent first and second complex valued distances between the first boundary and a first theoretical image plane and the second boundary and a second theoretical image plane, h 1 and h 2 represent distances between the directional resistivity tool and the first and second boundaries, and k n is the wave number of the first formation layer.
14. The method of claim 9 , wherein the at least one analytical expression comprises an expression relating at least one cross-coupling component to at least the first formation parameter, the expression being selected from at least one of the following:
H
ZX
=
M
4
π
{
A
1
(
2
d
1
+
2
h
1
)
a
sin
(
θ
-
β
1
)
+
β
1
sin
(
θ
-
β
1
)
cos
(
θ
-
β
1
)
︸
from
image
source
1
+
M
4
π
{
-
A
2
(
2
d
2
+
2
h
2
)
a
sin
(
θ
+
β
2
)
+
B
2
sin
(
θ
+
β
2
)
cos
(
θ
+
β
2
)
︸
from
image
source
2
H
XZ
=
M
X
4
π
{
-
A
1
(
2
d
1
+
2
h
1
)
a
sin
(
θ
-
β
1
)
+
B
1
sin
(
θ
-
β
1
)
cos
(
θ
-
β
1
)
︸
from
image
source
1
+
M
X
4
π
{
A
2
(
2
d
2
+
2
h
2
)
a
sin
(
θ
+
β
2
)
+
β
2
sin
(
θ
+
β
2
)
cos
(
θ
+
β
2
)
︸
from
image
source
2
wherein
A
1
=
ⅇ
-
j
k
n
r
1
r
1
5
(
k
n
2
r
1
2
-
3
j
k
n
r
1
-
3
)
,
A
2
=
ⅇ
-
j
k
n
r
2
r
2
5
(
k
n
2
r
2
2
-
3
j
k
n
r
2
-
3
)
B
1
=
ⅇ
-
j
k
n
r
1
r
1
3
(
k
n
2
r
1
2
+
j
k
n
r
1
+
1
)
,
B
2
=
ⅇ
-
j
k
n
r
2
r
2
3
(
k
n
2
r
2
2
+
j
k
n
r
2
+
1
)
and wherein H ZX and H XZ the cross-coupling components, M Z and M X represent the dipole moments of z-mode and x-mode transmitters, a represents an axial distance between a transmitter and a receiver, r 1 and r 2 represent distances between first and second image sources and the receiver, d 1 and d 2 represent first and second complex valued distances between the first boundary and a first theoretical image plane and the second boundary and a second theoretical image plane, h 1 and h 2 represent distances between the directional resistivity tool and the first and second boundaries, k n is the wave number of the first formation layer, θ represents a dip angle of the directional resistivity with respect to a horizontal plane, β 1 represents a dip angle of the first boundary with respect to the horizontal plane, and β 2 represents a dip angle of the second boundary with respect to the horizontal plane.
15. The method of claim 9 , wherein the at least one analytical expression comprises an expression relating at least one coupling component to at least the first formation parameter, the expression being selected from at least one of the following:
H
ZZ
=
M
Z
4
π
{
A
1
(
2
d
1
+
2
h
1
)
2
-
B
1
cos
2
(
θ
-
β
1
)
+
2
ⅇ
-
j
k
n
r
1
r
1
3
(
j
k
n
r
1
+
1
)
}
︸
from
image
source
1
+
M
Z
4
π
{
A
2
(
2
d
2
+
2
h
2
)
2
-
B
2
cos
2
(
θ
+
β
2
)
+
2
ⅇ
-
j
k
n
r
2
r
2
3
(
j
k
n
r
2
+
1
)
}
︸
from
image
source
2
+
M
Z
4
π
2
ⅇ
-
j
k
n
a
a
3
(
j
k
n
a
+
1
)
︸
from
real
source
H
XX
=
M
X
4
π
{
A
1
(
2
d
1
+
2
h
1
)
2
-
B
1
sin
2
(
θ
-
β
1
)
+
2
ⅇ
-
j
k
n
r
1
r
1
3
(
j
k
n
r
1
+
1
)
}
︸
from
image
source
1
+
M
X
4
π
{
A
2
(
2
d
2
+
2
h
2
)
2
-
B
2
sin
2
(
θ
+
β
2
)
+
2
ⅇ
-
j
k
n
r
2
r
2
3
(
j
k
n
r
2
+
1
)
}
︸
from
image
source
2
+
M
X
4
π
ⅇ
-
j
k
n
a
a
3
(
k
n
2
a
2
-
j
k
n
a
-
1
)
︸
from
real
source
wherein
A
1
=
ⅇ
-
j
k
n
r
1
r
1
5
(
k
n
2
r
1
2
-
3
j
k
n
r
1
-
3
)
,
A
2
=
ⅇ
-
j
k
n
r
2
r
2
5
(
k
n
2
r
2
2
-
3
j
k
n
r
2
-
3
)
B
1
=
ⅇ
-
j
k
n
r
1
r
1
3
(
k
n
2
r
1
2
+
j
k
n
r
1
+
1
)
,
B
2
=
ⅇ
-
j
k
n
r
2
r
2
3
(
k
n
2
r
2
2
+
j
k
n
r
2
+
1
)
and wherein H ZZ and H XX the coupling components, M Z and M X represent the dipole moments of z-mode and x-mode transmitters, a represents an axial distance between a transmitter and a receiver, r 1 and r 2 represent distances between first and second image sources and the receiver, d 1 and d 2 represent first and second complex valued distances between the first boundary and a first theoretical image plane and the second boundary and a second theoretical image plane, h 1 and h 2 represent distances between the directional resistivity tool and the first and second boundaries, k n is the wave number of the first formation layer, θ represents a dip angle of the directional resistivity with respect to a horizontal plane, β 1 represents a dip angle of the first boundary with respect to the horizontal plane, and β 2 represents a dip angle of the second boundary with respect to the horizontal plane.
16. A method for estimating a value of at least a first formation parameter from a directional resistivity measurement, the method comprising:
(a) acquiring a forward model of a subterranean formation, the forward model including at least one analytical expression relating a directional resistivity measurement to at least the first formation parameter, the analytical expression including at least one image source term;
(b) causing a downhole processor to compute a plurality of hypothetical directional resistivity measurements for a corresponding plurality of values of the first formation parameters;
(c) acquiring at least one directional resistivity measurement;
(d) selecting a value of at least the first formation parameter for which the hypothetical directional resistivity measurement computed in (b) substantially matches the directional resistivity measurement acquired in (c).
17. The method of claim 16 , wherein the forward model acquired in (a) comprises a two-layer formation model in which a directional resistivity tool is deployed in a first formation layer spaced apart from a boundary between the first formation layer and a second formation layer.
18. The method of claim 16 , wherein the directional resistivity tool is substantially parallel with the boundary between the first and second formation layers.
19. The method of claim 16 , wherein the forward model acquired in (a) comprises a three-layer model in which a directional resistivity tool is deployed in a first formation layer between first and second boundaries, the first boundary between the first formation layer and a second formation layer and the second boundary between the first formation layer and a third formation layer.
20. The method of claim 19 , wherein the first boundary, the second boundary, and the directional resistivity tool are substantially parallel with one another.