# US 10,502,859 B2 · Method and apparatus for determining resistivity of a formation

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- Number: US 10,502,859 B2
- Document ID: US10502859B2
- Status: Granted
- Kind: B2
- Application number: 15/502,929
- Filed: 2015-08-06
- Published: 2017-08-10
- Granted: 2019-12-10
- Earliest claimed: 2014-08-11
- Assignees: Schlumberger Technology Corporation
- Applicants: Schlumberger Technology Corporation
- Inventors: Emmanuel Legendre; Alexandre Abellan
- Web page: https://patentgrep.com/patent/US10502859B2

## CPC

- G01V 3/20: operating with propagation of electric current
- G01V 3/24: using AC
- G01V 3/38: Processing data, e.g. for analysis, for interpretation, for correction
- G01V: GEOPHYSICS; GRAVITATIONAL MEASUREMENTS; DETECTING MASSES OR OBJECTS; TAGS (means for indicating the location of accidentally buried, e.g. snow-buried, persons A63B29/02)

## Abstract

Apparatus and methods operable to determine a resistivity of a subterranean formation surrounding a wellbore. One such method includes using an apparent impedance function depending on a frequency variable and a plurality of unknown parameters, at least one of the unknown parameters depending on a formation impedance of the subterranean formation. The method also includes applying a voltage, at each of a plurality of frequency values, between electrodes of a resistivity tool that is disposed in the wellbore. The method also includes measuring, across the electrodes, a plurality of apparent impedance values, each corresponding to a different one of the frequency values. The method still further includes determining the unknown parameters based on the frequency values and the apparent impedance values, and estimating the formation resistivity based on an expression that includes at least one of the unknown parameters.

## Claims

### Claim 1

A method for estimating a formation resistivity of a subterranean formation, comprising:
using an apparent impedance function (Zapp) depending on a frequency variable (ω) and on a plurality of unknown parameters, at least one of the unknown parameters depending on a formation impedance (Zform) of the subterranean formation, so that the apparent impedance function (Zapp) is non-linear relative to the formation impedance (Zform);
applying a voltage, at each of a plurality of frequency values (ω1, ω2, ω3), between electrodes of a resistivity that is disposed in a wellbore proximate the subterranean formation penetrated by the wellbore;
measuring, across the electrodes, a plurality of apparent impedance values (Zapp1, Zapp2, Zapp3), each corresponding to a different one of the plurality of frequency values (ω1, ω2, ω3);
determining the plurality of unknown parameters based on the plurality of frequency values (ω1, ω2, ω3) and the plurality of apparent impedance values (Zapp1, Zapp2, Zapp3); and
estimating the formation resistivity based on an expression that includes at least one of the plurality of unknown parameters, wherein:
the apparent impedance function (Zapp) when multiplied by the frequency variable (ω) corresponds to a ratio of polynomials depending on the frequency variable (ω) and having the plurality of unknown parameters as coefficients, such as:
Z
app
·
ω
=
∑
i
=
0
p
⁢
a
i
·
ω
i
1
+
∑
i
=
1
q
⁢
b
i
·
ω
i
wherein Zapp is the apparent frequency function,
ω is the frequency variable, and
ai and bi are the unknown parameters.

### Claim 2 (depends on 1)

The method of claim 1, wherein the expression is a1−a0·b1.

### Claim 3 (depends on 1)

The method of claim 1, wherein estimating the formation resistivity comprises determining a modulus or a real part of the expression.

### Claim 4 (depends on 1)

The method of claim 1, wherein the apparent impedance function (Zapp) is a non-linear relative to the frequency variable (ω).

### Claim 5 (depends on 1)

The method of claim 1, wherein the plurality of apparent impedance values (Zapp1, Zapp2, Zapp3) comprises at least three apparent impedance values and the plurality of unknown parameters comprise at least three unknown parameters.

### Claim 6 (depends on 1)

The method of claim 1, wherein measuring the plurality of apparent impedance values (Zapp1, Zapp2, Zapp3) comprises sensing a ratio of the voltage to a corresponding one of a plurality of currents generated at the corresponding one of a plurality of frequency values (ω1, ω2, ω3), each of the plurality of currents flowing along a corresponding one of a plurality of conductive paths between the electrodes.

### Claim 7 (depends on 1)

The method of claim 1, wherein at least one of the plurality of conductive paths extends at least partially through oil-based drilling fluid in which the resistivity tool is at least partially submersed within the wellbore.

### Claim 8 (depends on 1)

The method of claim 1, wherein the unknown parameters are complex parameters.

### Claim 9

A method for estimating a formation resistivity of a subterranean formation, comprising:
using an apparent impedance function (Zapp) depending on a frequency variable (ω) and on a plurality of unknown parameters, at least one of the unknown parameters depending on a formation impedance (Zform) of the subterranean formation, so that the apparent impedance function (Zapp) is non-linear relative to the formation impedance (Zform);
applying a voltage, at each of a plurality of frequency values (ω1, ω2, ω3), between electrodes of a resistivity that is disposed in a wellbore proximate the subterranean formation penetrated by the wellbore;
measuring, across the electrodes, a plurality of apparent impedance values (Zapp1, Zapp2, Zapp3), each corresponding to a different one of the plurality of frequency values (ω1, ω2, ω3);
determining the plurality of unknown parameters based on the plurality of frequency values (ω1, ω2, ω3) and the plurality of apparent impedance values (Zapp1, Zapp2, Zapp3); and
estimating the formation resistivity based on an expression that includes at least one of the plurality of unknown parameters, wherein the expression is obtained by applying:
lim
ω
→
0
⁢
(
∂
(
Z
app
·
ω
)
∂
ω
)
,
wherein Zapp is the apparent frequency function and co is the frequency variable.

### Claim 10 (depends on 9)

The method of claim 9, wherein measuring the plurality of apparent impedance values (Zapp1, Zapp2, Zapp3) comprises sensing a ratio of the voltage to a corresponding one of a plurality of currents generated at the corresponding one of a plurality of frequency values (ω1, ω2, ω3), each of the plurality of currents flowing along a corresponding one of a plurality of conductive paths between the electrodes.

### Claim 11 (depends on 10)

The method of claim 10, wherein at least one of the plurality of conductive paths extends at least partially through oil-based drilling fluid in which the resistivity tool is at least partially submersed within the wellbore.

### Claim 12 (depends on 9)

The method of claim 9, wherein estimating the formation resistivity comprises determining a modulus or a real part of the expression.

### Claim 13 (depends on 9)

The method of claim 9, wherein the apparent impedance function (Zapp) is a non-linear relative to the frequency variable (ω).

### Claim 14 (depends on 9)

The method of claim 9, wherein the plurality of apparent impedance values (Zapp1, Zapp2, Zapp3) comprises at least three apparent impedance values and the plurality of unknown parameters comprise at least three unknown parameters.

### Claim 15 (depends on 9)

The method of claim 9, wherein the unknown parameters are complex parameters.

### Claim 16

An apparatus for estimating a formation resistivity of a subterranean formation penetrated by a wellbore, comprising:
a resistivity tool comprising external electrodes and operable within the wellbore to:
apply a voltage, at each of a plurality of frequency values (ω1, ω2, ω3), between the electrodes; and
measure, across the electrodes, a plurality of apparent impedance values (Zapp1, Zapp2, Zapp3), each corresponding to a different one of the plurality of frequency values (ω1, ω2, ω3); and
a processing system, wherein at least one of the resistivity tool and the processing system is operable to:
use an apparent impedance function (Zapp) depending on a frequency variable (ω) and on a plurality of unknown parameters, at least one of the unknown parameters depending on a formation impedance (Zform) of the subterranean formation, so that the apparent impedance function (Zapp) is non-linear relative to the formation impedance (Zform),
determine the plurality of unknown parameters based on the plurality of frequency values (ω1, ω2, ω3) and the plurality of apparent impedance values (Zapp1, Zapp2, Zapp3); and
estimate the formation resistivity based on an expression that includes at least one of the plurality of unknown parameters, wherein the expression is obtained by applying:
lim
ω
→
0
⁢
(
∂
(
Z
app
·
ω
)
∂
ω
)
,
wherein Zapp is the apparent frequency function and co is the frequency variable.

### Claim 17 (depends on 16)

The apparatus of claim 16, wherein the resistivity tool further comprises the processing system.

### Claim 18 (depends on 16)

The apparatus of claim 16, further comprising surface equipment disposed at a wellsite surface associated with the wellbore, wherein the surface equipment comprises at least a portion of the processing system.

### Claim 19

A computer program, comprising machine-readable instructions for:
using an apparent impedance function (Zapp) depending on a frequency variable (ω) and on a plurality of unknown parameters, at least one of the unknown parameters depending on a formation impedance (Zform) of the subterranean formation, so that the apparent impedance function (Zapp) is non-linear relative to the formation impedance (Zform),
on the basis of a plurality of measured apparent impedance values (Zapp1, Zapp2, Zapp3) each corresponding to a different one of a plurality of frequency values (ω1, ω2, ω3), determining the plurality of unknown parameters;
estimating the formation resistivity based on an expression that includes at least one of the plurality of unknown parameters, wherein:
the apparent impedance function (Zapp) when multiplied by the frequency variable (ω) corresponds to a ratio of polynomials depending on the frequency variable (ω) and having the plurality of unknown parameters as coefficients, such as:
Z
app
·
ω
=
∑
i
=
0
p
⁢
a
i
·
w
i
1
+
∑
i
=
1
q
⁢
b
i
·
ω
1
.

### Claim 20 (depends on 19)

A computer readable storage medium storing the computer program of claim 19.