Method and apparatus for determining resistivity of a formation

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.

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.

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
• .