Method to estimate formation resistivity
A method and system for estimating a resistivity of a formation. A method for estimating a resistivity of a formation may comprise disposing a downhole tool into a borehole, wherein the downhole tool comprises a pad, an injector electrode, and a return electrode, injecting a current signal into the formation from the injector electrode, measuring a voltage signal between the injector electrode and the return electrode; and determining a formation resistivity and a formation dielectric constant from at least one of the voltage signal, at least one property of the downhole tool, and at least one property of the borehole. A system for estimating a resistivity of a formation may comprise a downhole tool. The downhole tool may comprise a pad, wherein the pad comprises an injector electrode and a return electrode. The system may further comprise a conveyance for disposing the downhole tool in a borehole and an information handling system.
1. A method for estimating a resistivity of a formation, comprising:
disposing a downhole tool into a borehole, wherein the downhole tool comprises a pad, an injector electrode, and a return electrode;
injecting current signals into the formation from the injector electrode at a first frequency and a second frequency;
measuring voltage signals between the injector electrode and the return electrode at the first frequency and the second frequency;
calculating a first impedance at the first frequency and calculating a second impedance at the second frequency;
determining the first impedance at the first frequency using
Z
1
=
(
ρ
M
1
h
A
)
[
1
-
j
[
2
π
ɛ
0
(
ρ
M
1
f
1
k
M
1
)
]
1
+
[
2
π
ɛ
0
(
ρ
M
1
f
1
k
M
1
)
]
2
]
+
(
ρ
F
d
A
)
[
1
-
j
[
2
π
ɛ
0
(
ρ
F
f
1
k
F
)
]
1
+
[
2
π
ɛ
0
(
ρ
F
f
1
k
F
)
]
2
]
and determining the second impedance at the second frequency using
Z
2
=
(
ρ
M
2
h
A
)
[
1
-
j
[
2
π
ɛ
0
(
ρ
M
2
f
2
k
M
2
)
]
1
+
[
2
π
ɛ
0
(
ρ
M
2
f
2
k
M
2
)
]
2
]
+
(
ρ
F
d
A
)
[
1
-
j
[
2
π
ɛ
0
(
ρ
F
f
2
k
F
)
]
1
+
[
2
π
ɛ
0
(
ρ
F
f
2
k
F
)
]
2
]
.
wherein ρ M1 and ρ M2 denote the mud resistivity at frequencies f 1 and f 2 respectively, k M1 and k M2 denote the mud dielectric constant at frequencies f 1 and f 2 respectively, ρ F denotes the formation resistivity, k F denotes the formation dielectric constant, h denotes the standoff, A denotes the area of the injector electrode, d denotes the effective depth of investigation, and ε 0 is the dielectric permittivity of the vacuum; and
determining both a formation resistivity and a formation dielectric constant from the voltage signals at the first frequency and at the second frequency.
2. The method of claim 1 , wherein determining both the formation resistivity and the formation dielectric constant from the voltage signals at the first frequency and at the second frequency, further comprises from at least one property of the downhole tool, and at least one property of the borehole.
3. The method of claim 1 , wherein the at least one property of the borehole is a mud impedance, a formation impedance, a mud resistance, a formation resistance, a mud capacitance, or a formation capacitance.
4. The method of claim 1 , further comprising calculating a standoff of the injector electrode and identifying a characteristic of a borehole geometry from the standoff.
5. The method of claim 1 , further comprising performing a statistical measurement of the plurality of formation resistivities.
6. The method of claim 1 , further comprising performing a statistical measurement of the plurality of formation dielectric constants.
7. A method for estimating a resistivity of a formation, comprising:
disposing a down hole tool into a borehole, wherein the downhole tool comprises a pad, an injector electrode, and a return electrode;
injecting a voltage signal between the injector electrode and the return electrode at a first frequency and a second frequency;
measuring a current signal between the injector electrode and the return electrode at a first frequency and a second frequency;
calculating a first impedance at a first frequency and calculating a second impedance at a second frequency;
determining the first impedance at the first frequency using
Z
1
=
(
ρ
M
1
h
A
)
[
1
-
j
[
2
π
ε
0
(
ρ
M
1
f
1
k
M
1
)
]
1
+
[
2
π
ε
0
(
ρ
M
1
f
1
k
M
1
)
]
2
]
+
(
ρ
F
d
A
)
[
1
-
j
[
2
π
ε
0
(
ρ
F
f
1
k
F
)
]
1
+
[
2
π
ε
0
(
ρ
F
f
1
k
F
)
]
2
]
and determining the second impedance at the second frequency using
Z
2
=
(
ρ
M
2
h
A
)
[
1
-
j
[
2
π
ε
0
(
ρ
M
2
f
2
k
M
2
)
]
1
+
[
2
π
ε
0
(
ρ
M
2
f
2
k
M
2
)
]
2
]
+
(
ρ
F
d
A
)
[
1
-
j
[
2
π
ε
0
(
ρ
F
f
2
k
F
)
]
1
+
[
2
π
ε
0
(
ρ
F
f
2
k
F
)
]
2
]
,
wherein ρ M1 and ρ M2 denote the mud resistivity at frequencies f 1 and f 2 respectively, k M1 and k M2 denote the mud dielectric constant at frequencies f 1 and f 2 respectively, ρ F denotes the formation resistivity, k F denotes the formation dielectric constant, h denotes the standoff, A denotes the area of the injector electrode, d denotes the effective depth of investigation, and ε 0 is the dielectric permittivity of the vacuum; and
determining both a plurality of formation resistivities and a plurality of formation dielectric constants from the current signal at the first frequency and the second frequency.
8. The method of claim 7 , further comprising performing a statistical measurement of the plurality of formation resistivities.
9. The method of claim 8 , wherein the statistical measurement is a mean, median, or frequency weighted mean.
10. The method of claim 7 , further comprising performing a statistical measurement of the plurality of formation dielectric constants.
11. The method of claim 10 , wherein the statistical measurement is a mean, median, or frequency weighted mean.
12. The method of claim 7 , further comprising calculating a standoff of the injector electrode and identifying a characteristic of a borehole geometry from the standoff.
13. A system for estimating a resistivity of a formation, comprising:
a down hole tool, wherein the downhole tool comprises:
a pad, wherein the pad comprises an injector electrode and a return electrode; and
a conveyance for disposing the down hole tool in a borehole; and
an information handling system for:
determining a first impedance at a first frequency using
Z
1
=
(
ρ
M
1
h
A
)
[
1
-
j
[
2
π
ε
0
(
ρ
M
1
f
1
k
M
1
)
]
1
+
[
2
π
ε
0
(
ρ
M
1
f
1
k
M
1
)
]
2
]
+
(
ρ
F
d
A
)
[
1
-
j
[
2
π
ε
0
(
ρ
F
f
1
k
F
)
]
1
+
[
2
π
ε
0
(
ρ
F
f
1
k
F
)
]
2
]
and determining a second impedance at a second frequency using
Z
2
=
(
ρ
M
2
h
A
)
[
1
-
j
[
2
π
ε
0
(
ρ
M
2
f
2
k
M
2
)
]
1
+
[
2
π
ε
0
(
ρ
M
2
f
2
k
M
2
)
]
2
]
+
(
ρ
F
d
A
)
[
1
-
j
[
2
π
ε
0
(
ρ
F
f
2
k
F
)
]
1
+
[
2
π
ε
0
(
ρ
F
f
2
k
F
)
]
2
]
,
wherein ρ M1 and ρ M2 denote the mud resistivity at frequencies f 1 and f 2 respectively, k M1 and k M2 denote the mud dielectric constant at frequencies f 1 and f 2 respectively, ρ F denotes the formation resistivity, k F denotes the formation dielectric constant, h denotes the standoff, A denotes the area of the injector electrode, d denotes the effective depth of investigation, and ε 0 is the dielectric permittivity of the vacuum; and
determining both a formation resistivity and a formation dielectric constant from at least a voltage signal at a first frequency and a second frequency, at least one property of the downhole tool, and at least one property of the borehole.
14. The system of claim 13 , wherein the information handling system further identifies a first impedance from a first frequency.
15. The system of claim 14 , wherein the information handling system further identifies a second impedance from a second frequency.
16. The system of claim 15 , wherein the information handling system further performs a statistical measurement of the formation resistivity and wherein the statistical measurement is a mean, median, or frequency weighted mean.
17. The system of claim 15 , wherein the information handling system further performs a statistical measurement of the formation dielectric constant and wherein the statistical measurement is a mean, median, or frequency weighted mean.
18. The system of claim 13 , wherein the at least one property of the borehole is a mud impedance, a formation impedance, a mud resistance, a formation resistance, a mud capacitance, or a formation capacitance.
19. The system of claim 13 , wherein the at least one property of the downhole tool is an effective cross-sectional area of the injector electrode.
20. The system of claim 13 , wherein the information handling system further calculates a standoff of the injector electrode and identifies a characteristic of a borehole geometry from the standoff.
21. A method for estimating a resistivity of a formation, comprising:
disposing a down hole tool into a borehole, wherein the downhole tool comprises a pad, an injector electrode, and a return electrode;
injecting a current signal into the formation from the injector electrode;
measuring a voltage signal between the injector electrode and the return electrode;
calculating a first impedance at a first frequency and calculating a second impedance at a second frequency;
determining the first impedance at the first frequency using
Z
1
=
(
ρ
M
1
h
A
)
[
1
-
j
[
2
π
ε
0
(
ρ
M
1
f
1
k
M
1
)
]
1
+
[
2
π
ε
0
(
ρ
M
1
f
1
k
M
1
)
]
2
]
+
(
ρ
F
d
A
)
[
1
-
j
[
2
π
ε
0
(
ρ
F
f
1
k
F
)
]
1
+
[
2
π
ε
0
(
ρ
F
f
1
k
F
)
]
2
]
and determining the second impedance at the second frequency using
Z
2
=
(
ρ
M
2
h
A
)
[
1
-
j
[
2
π
ε
0
(
ρ
M
2
f
2
k
M
2
)
]
1
+
[
2
π
ε
0
(
ρ
M
2
f
2
k
M
2
)
]
2
]
+
(
ρ
F
d
A
)
[
1
-
j
[
2
π
ε
0
(
ρ
F
f
2
k
F
)
]
1
+
[
2
π
ε
0
(
ρ
F
f
2
k
F
)
]
2
]
,
wherein ρ M1 and ρ M2 denote the mud resistivity at frequencies f 1 and f 2 respectively, k M1 and k M2 denote the mud dielectric constant at frequencies f 1 and f 2 respectively, ρ F denotes the formation resistivity, k F denotes the formation dielectric constant, h denotes the standoff, A denotes the area of the injector electrode, d denotes the effective depth of investigation, and ε 0 is the dielectric permittivity of the vacuum;
determining a formation resistivity and a formation dielectric constant from at least one of the voltage signal, at least one property of the downhole tool, and at least one property of the borehole; and
calculating a standoff of the injector electrode and identifying a borehole characteristic of a borehole geometry from the standoff.
22. A system for estimating a resistivity of a formation, comprising:
a downhole tool, wherein the downhole tool comprises:
a pad, wherein the pad comprises an injector electrode and a return electrode; and
a conveyance for disposing the downhole tool in a borehole; and
an information handling system for:
determining a first impedance at a first frequency using
Z
1
=
(
ρ
M
1
h
A
)
[
1
-
j
[
2
π
ε
0
(
ρ
M
1
f
1
k
M
1
)
]
1
+
[
2
π
ε
0
(
ρ
M
1
f
1
k
M
1
)
]
2
]
+
(
ρ
F
d
A
)
[
1
-
j
[
2
π
ε
0
(
ρ
F
f
1
k
F
)
]
1
+
[
2
π
ε
0
(
ρ
F
f
1
k
F
)
]
2
]
and determining a second impedance at a second frequency using
Z
2
=
(
ρ
M
2
h
A
)
[
1
-
j
[
2
π
ε
0
(
ρ
M
2
f
2
k
M
2
)
]
1
+
[
2
π
ε
0
(
ρ
M
2
f
2
k
M
2
)
]
2
]
+
(
ρ
F
d
A
)
[
1
-
j
[
2
π
ε
0
(
ρ
F
f
2
k
F
)
]
1
+
[
2
π
ε
0
(
ρ
F
f
2
k
F
)
]
2
]
,
wherein ρ M1 and ρ M2 denote the mud resistivity at frequencies f 1 and f 2 respectively, k M1 and k M2 denote the mud dielectric constant at frequencies f 1 and f 2 respectively, ρ F denotes the formation resistivity, k F denotes the formation dielectric constant, h denotes the standoff, A denotes the area of the injector electrode, d denotes the effective depth of investigation, and ε 0 is the dielectric permittivity of the vacuum;
determining a formation resistivity and a formation dielectric constant from at least a voltage signal, at least one property of the downhole tool, and at least one property of the borehole; and
calculating a standoff of the injector electrode and identifying a borehole characteristic of a borehole geometry from the standoff.