Multi-frequency cancellation of dielectric effect

• US 7,714,585 B2
• Filed: 03/21/2007
• Issued: 05/11/2010
• Est. Priority Date: 03/21/2007
• Status: Active Grant

• Current Assignee: Baker Hughes Incorporated (Baker Hughes Company)
• Original Assignee: Baker Hughes Incorporated (Baker Hughes Company)
• Inventors: Nikitenko, Marina N., Tabarovsky, Leonty A.
• Primary Examiner(s): Assouad; Patrick J
• Assistant Examiner(s): Schindler; David M.
• Application Number:
  US11/689,118
• Publication Number:
  US 20080231283A1
• Time in Patent Office: 1,147 Days
• Field of Search: None
• US Class Current: 324/338
• CPC Class Codes: G01V 3/24 using ac

Abstract

Measurements made with an induction logging tool are responsive to formation conductivity and permittivity. The effect of permittivity can be substantially removed by multifrequency focusing.

Claims

1. A method of determining a resistivity of an earth formation, the method comprising:

(a) conveying into a borehole in the earth formation a resistivity measuring instrument having at least one transmitter and at least one receiver spaced apart from the at least one transmitter;

(b) activating the at least one transmitter at a plurality of frequencies and inducing signals in the at least one receiver, the induced signals indicative of the resistivity and a permittivity of the earth formation; and

(c) applying a multifrequency focusing (MFF) to the induced signals by forming a weighted summation thereof to give a focused signal from which the resistivity is determined;

wherein coefficients of the MFF are selected to substantially eliminate an effect of the permittivity on the focused signal.

2. The method of claim 1 wherein forming the weighted summation further comprises using a relation of the form

$H_f=\sum _{}^{}$

j = 1 N ⁢

⁢

M j ·

H ⁡

( f j )

where H(f_j) refers to the magnetic field at the j-th frequency and M_j are coefficients selected using a relationship of the form;

$\left[\begin{array}{ccccc}1& 1& 1& \mathrm{\dots <br><br>}& 1\\ {\left(\frac{f_2}{f_1}\right)}^{\frac{1}{2}}& {\left(\frac{f_3}{f_1}\right)}^{\frac{1}{2}}& {\left(\frac{f_4}{f_1}\right)}^{\frac{1}{2}}& \mathrm{\dots <br><br>}& {\left(\frac{f_N}{f_1}\right)}^{\frac{1}{2}}\\ \frac{f_2}{f_1}& \frac{f_3}{f_1}& \frac{f_4}{f_1}& \mathrm{\dots <br><br>}& \frac{f_N}{f_1}\\ \mathrm{\dots <br><br>}& \mathrm{\dots <br><br>}& \mathrm{\dots <br><br>}& \mathrm{\dots <br><br>}& \mathrm{\dots <br><br>}\\ {\left(\frac{f_2}{f_1}\right)}^{\frac{N-3}{2}}& {\left(\frac{f_3}{f_1}\right)}^{\frac{N-3}{2}}& {\left(\frac{f_4}{f_1}\right)}^{\frac{N-3}{2}}& \mathrm{\dots <br><br>}& {\left(\frac{f_N}{f_1}\right)}^{\frac{N-3}{2}}\end{array}\right]<br><br>\left[\begin{array}{c}{M}_2^{\mathrm{\prime <br><br>}}\\ M_3^{\mathrm{\prime <br><br>}}\\ M_4^{\mathrm{\prime <br><br>}}\\ \mathrm{\dots <br><br>}\\ M_N^{\mathrm{\prime <br><br>}}\end{array}\right]=-\left[\begin{array}{c}1\\ 1\\ 1\\ \mathrm{\dots <br><br>}\\ 1\end{array}\right]$

wherein

$M_i=\frac{f_1^2}{f_i^2}<br><br>M_i^{\mathrm{\prime <br><br>}},$

M₁^′

here M_i s are the coefficients and the f'"'"'s are frequencies.

3. The method of claim 2 wherein using signals at the at least one receiver further comprises using signals at a main receiver and a second receiver wherein the at least one transmitter, the main receiver and the second receiver define a three-coil array.

4. The method of claim 3 further comprising using a relationship of the form:

$\frac{\mathrm{\mu <br><br>}·<br><br>\mathrm{\pi <br><br>}·<br><br>f^2}{L}<br><br>\{\left(1-d^2\right)+L·<br><br>\left(1-d^3\right)·<br><br>c^{\frac{1}{2}}<br><br>f^{\frac{1}{2}}+L^2·<br><br>\left(1-d^4\right)·<br><br>\frac{\mathrm{cf}}{2!}+L^3·<br><br>\left(1-d^5\right)·<br><br>\frac{c^{\frac{3}{2}}<br><br>f^{\frac{3}{2}}}{3!}+L^4·<br><br>\left(1-d^6\right)·<br><br>\frac{c^2<br><br>f^2}{4!}+\mathrm{\dots <br><br>}<br><br>\}$

where μ

is a permeability, L is a distance between the at least one transmitter transmitter and the main receiver, f is a frequency, d is a ratio of a smaller length to a larger length of the three coil array, and c=−

2π

iμ

σ

, where μ

is a permeability and σ

is a conductivity.

5. The method of claim 1 further comprising conveying the resistivity instrument into the borehole on one of:

(i) a wireline, (ii) a slickline, and (iii) a drilling tubular.

6. The method of claim 1 further comprising using the determined resistivity for performing an operation that is at least one of (i) displaying the formation resistivity, (ii) storing the formation resistivity on a suitable medium, (iii) interpreting a petrophysical property of the formation, (iv) performing a geological correlation, (v) determining a hydrocarbon-water contact, (vi) determining an invasion profile, (vii) identifying a fracture, (viii) estimating movable hydrocarbons, (ix) controlling a directional drilling of the borehole, and (x) drilling an additional well.

7. That method of claim 1 further comprising selecting values of the plurality of frequencies to increase linear independence of columns of a Taylor series matrix.

8. An apparatus configured for determining a resistivity of an earth formation, the apparatus comprising:

a resistivity measuring instrument configured to be conveyed into a borehole in the earth formation, the resistivity instrument having at least one transmitter and at least one receiver spaced apart from the at least one transmitter, the at least one transmitter configured to operate at a plurality of frequencies, the at least one receiver configured to produce induced signals indicative of the resistivity and a permittivity of the earth formation; and

a processor configured to apply a multifrequency focusing (MFF) to the induced signals by forming a weighted summation thereof to give a focused signal from which the resistivity is determined using coefficients of the MFF selected to substantially eliminate an effect of the permittivity on the focused signal.

9. The apparatus of claim 8 wherein the processor is further configured to form the weighted summation by using a relation of the form:

$H_f=\underset{j=1}{\overset{N}{\sum <br><br>}}<br><br><br><br>M_j·<br><br>H<br><br>\left(f_j\right)$

where H(f_j) refers to the magnetic field at the j-th frequency and M_j are coefficients selected using a relationship of the form;

$\left[\begin{array}{ccccc}1& 1& 1& \mathrm{\dots <br><br>}& 1\\ {\left(\frac{f_2}{f_1}\right)}^{\frac{1}{2}}& {\left(\frac{f_3}{f_1}\right)}^{\frac{1}{2}}& {\left(\frac{f_4}{f_1}\right)}^{\frac{1}{2}}& \mathrm{\dots <br><br>}& {\left(\frac{f_N}{f_1}\right)}^{\frac{1}{2}}\\ \frac{f_2}{f_1}& \frac{f_3}{f_1}& \frac{f_4}{f_1}& \mathrm{\dots <br><br>}& \frac{f_N}{f_1}\\ \mathrm{\dots <br><br>}& \mathrm{\dots <br><br>}& \mathrm{\dots <br><br>}& \mathrm{\dots <br><br>}& \mathrm{\dots <br><br>}\\ {\left(\frac{f_2}{f_1}\right)}^{\frac{N-3}{2}}& {\left(\frac{f_3}{f_1}\right)}^{\frac{N-3}{2}}& {\left(\frac{f_4}{f_1}\right)}^{\frac{N-3}{2}}& \mathrm{\dots <br><br>}& {\left(\frac{f_N}{f_1}\right)}^{\frac{N-3}{2}}\end{array}\right]<br><br>\left[\begin{array}{c}{M}_2^{\mathrm{\prime <br><br>}}\\ M_3^{\mathrm{\prime <br><br>}}\\ M_4^{\mathrm{\prime <br><br>}}\\ \mathrm{\dots <br><br>}\\ M_N^{\mathrm{\prime <br><br>}}\end{array}\right]=-\left[\begin{array}{c}1\\ 1\\ 1\\ \mathrm{\dots <br><br>}\\ 1\end{array}\right]$

wherein

$M_i=\frac{f_1^2}{f_i^2}<br><br>M_i^{\mathrm{\prime <br><br>}},$

M₁^′

here M_i s are the coefficients and the f'"'"'s are frequencies.

10. The apparatus of claim 9 wherein the at least one receiver further comprises a main receiver and a second receiver,

wherein the at least one transmitter, the main receiver and the second receiver defining a three-coil array.

11. The apparatus of claim 10 wherein the processor is further configured to use a relationship of the form

$\frac{\mu }{}$

·

π

·

f 2 L ⁢

{ ( 1 - d 2 ) + L ·

( 1 - d 3 ) ·

c 1 2 ⁢

f 1 2 + L 2 ·

( 1 - d 4 ) ·

cf 2 ! + L 3 ·

( 1 - d 5 ) ·

c 3 2 ⁢

f 3 2 3 ! + L 4 ·

( 1 - d 6 ) ·

c 2 ⁢

f 2 4 ! + …

⁢

}

where μ

is a permeability, L is a distance between the at least one transmitter transmitter and the main receiver, f is a frequency, d is a ratio of a smaller length to a larger length of the three coil array, and c=−

2π

iμ

σ

, where μ

is a permeability and σ

is a conductivity.

12. The apparatus of claim 8 further comprising a conveyance device configured to convey the resistivity instrument into the borehole, the conveyance device selected from:

(i) a wireline, (ii) a slickline, and (iii) a drilling tubular.

13. The apparatus of claim 8 wherein a processor is further configured to perform at least one of (i) displaying the formation resistivity, (ii) storing the formation resistivity on a suitable medium, (iii) interpreting a petrophysical property of the formation, (iv) performing a geological correlation, (v) determining a hydrocarbon-water contact, (vi) determining an invasion profile, (vii) identifying a fracture, (viii) estimating movable hydrocarbons, (ix) controlling a directional drilling of the borehole, and (x) drilling an additional well.

14. The method of claim 8 wherein the processor is further configured to select values of the plurality of frequencies to increase linear independence of columns of a Taylor series matrix.

15. A computer readable medium product having stored thereon instructions that when read by a processor cause the processor to execute a method, the method comprising:

applying a multifrequency focusing (MFF) to signals induced in a receiver in a borehole in an earth formation at a plurality of frequencies by forming a weighted summation thereof to give a focused signal, the induced signals being indicative of a resistivity and a permittivity of the formation;

selecting coefficients of the MFF to substantially eliminate an effect of the permittivity on the focused signal; and

determining from the focused signal the formation resistivity.

16. The computer-readable medium of claim 15 further comprising at least one of:

(i) a read-only memory (ROM), (ii) a programmable read-only memory (PROM), (iii) an electrically programmable read-only memory (EPROM), (iv) an electrically alterable read-only memory (EAROM), (v) an electrically erasable and programmable read-only memory (EEPROM), (vi) a flash memory, (vii) an optical disk, (viii) a hard drive, and (ix) a non-volatile read-write memory (NOVRAM).