Systems and methods for determining dielectric constant or resistivity from electromagnetic propagation measurement using contraction mapping

• US 10,450,861 B2
• Filed: 12/02/2015
• Issued: 10/22/2019
• Est. Priority Date: 12/16/2014
• Status: Active Grant

• Current Assignee: Schlumberger Technology Corporation (Schlumberger NV)
• Original Assignee: Schlumberger Technology Corporation (Schlumberger NV)
• Inventors: Wang, Gong Li, Zhao, Tianxia, Sun, Keli, Abubakar, Aria
• Primary Examiner(s): Nghiem, Michael P
• Assistant Examiner(s): Ngo, Dacthang P
• Application Number:
  US14/957,531
• Publication Number:
  US 20160170069A1
• Time in Patent Office: 1,420 Days
• Field of Search: 702 7
• CPC Class Codes: E21B 49/00 Testing the nature of borehole walls; Formation testing; Methods or apparatus for obtaining samples of soil or well fluids, specially adapted to earth drilling or wells
• CPC Class Codes: G01V 3/28 using induction coils
• CPC Class Codes: G01V 3/38 Processing data, e.g. for analysis, for interpretation, for correction

Abstract

Identifying the dielectric constant and/or the electrical resistivity of part of a geological formation may reveal useful characteristics of the geological formation. This disclosure provides methods, systems, and machine-readable media to determine dielectric constant or electrical resistivity, or both, using contraction mapping. Specifically, contraction mapping may be used with a function of wavenumber k to iteratively solve for values of dielectric constant and electrical resistivity until convergence is achieved. This may allow for convergence to a solution without computing partial derivatives and/or with fewer iterations than previous techniques.

Claims

1. A method comprising:

placing a downhole tool in a wellbore in a geological formation;

using the downhole tool to emit a signal from a transmitter into the geological formation and receive the signal by a near receiver and a far receiver to obtain an electromagnetic attenuation measurement and an electromagnetic phase shift measurement; and

with a processor, solving for dielectric constant or electrical resistivity, or both, via contraction mapping to iteratively solve for wavenumber using;

distances of the near and far receivers to the transmitter; and

a function of wavenumber having attenuation and phase shift as a complex constant.

2. The method of claim 1, wherein using the downhole tool to obtain the electromagnetic attenuation measurement and the electromagnetic phase shift measurement comprises detecting the signal from the near receiver a first distance from the transmitter and detecting the signal from the far receiver a second distance from the transmitter, wherein the first distance is less than the second distance.

3. The method of claim 2, wherein the function of wavenumber accords with the following relationships:

4. The method of claim 2, wherein the function of wavenumber accords with the following relationships:

5. The method of claim 3, wherein the frequency of the signal that is emitted by the transmitter is between 50 kHz and 5 MHz.

6. The method of claim 3, wherein a spacing between the transmitter and a midpoint between the near receiver and the far receiver is between 16 inches and 40 inches.

7. The method of claim 4, wherein the frequency of the signal that is emitted by the first transmitter is between 50 kHz and 5 MHz.

8. The method of claim 4, wherein a spacing between the transmitter and a midpoint between the near receiver and the far receiver is between 16 inches and 40 inches.

9. The method of claim 1, wherein iteratively solving for wavenumber comprises starting with an initial guess of dielectric constant as a component of wavenumber, wherein the initial guess for the dielectric constant is a value between 0 and 10,000.

10. The method of claim 1, wherein iteratively solving for wavenumber comprises starting with an initial guess of electrical resistivity as a component of wavenumber, wherein the initial guess for the electrical resistivity in units of ohm-meters is a value between 0 and 1000.

11. A system comprising:

a transmitter configured to emit an electromagnetic signal of a first frequency into a geological formation;

a near receiver configured to receive the electromagnetic signal;

a far receiver configured to receive the electromagnetic signal, wherein the far receiver is spaced farther from the transmitter than the near receiver; and

a processor configured to use contraction mapping to iteratively solve for a dielectric constant and an electrical resistivity of the geological formation using a function of wavenumber having an attenuation of the electromagnetic signal between the near receiver and the far receiver and based on a phase shift of the electromagnetic signal between the near receiver and the far receiver as a complex constant.

12. The system of claim 11, wherein the function of wavenumber comprises a first-order term k(r_far−

r_near) but not a second-order term ½

ik²(r_near²−

r_far²);

wherein k represents wavenumber, r_far represents a relative position of the far receiver, and r_near represents a relative position of the near receiver.

13. The system of claim 11, wherein the function of wavenumber comprises a second-order term ½

ik²(r_near²−

r_far²) but not a first-order term k(r_far−

r_near);

wherein k represents wavenumber, r_far represents a relative position of the far receiver, and r_near represents a relative position of the near receiver.

14. The system of claim 11, wherein the processor is configured to use a different one of a plurality of functions of wavenumber depending on expected values of electrical resistivity and dielectric constant.

15. The system of claim 14, wherein the processor is configured to use a first of the plurality of functions of wavenumber in a relatively higher resistivity and lower dielectric constant region and to use a second of the plurality of functions of wavenumber in a relatively lower resistivity and higher dielectric constant region.

16. The system of claim 15, wherein the first of the plurality of functions of wavenumber accords with the following relationship:

17. One or more non-transitory machine-readable media comprising instructions to:

receive an attenuation measurement and a phase-shift measurement of a geological formation obtained by an electromagnetic downhole tool at a depth of the geological formation in a wellbore, wherein the electromagnetic downhole tool comprises a transmitter for emitting a signal into the geological formation and first and second receivers for receiving the signal; and

without computing any partial derivatives, use contraction mapping to iteratively solve for a dielectric constant and a resistivity of the geological formation at the depth based on distances of the first and second receivers to the transmitter and the attenuation measurement and the phase-shift measurement until the solution for dielectric constant converges and the solution for resistivity converges.

18. The one or more non-transitory machine-readable media of claim 17, wherein the instructions to use contraction mapping comprise instructions to use contraction mapping with a function of wavenumber k having a complex constant PS+iAT, wherein AT represents attenuation and PS represents phase shift.

19. The one or more non-transitory machine-readable media of claim 18, wherein the instructions to use contraction mapping to iteratively solve for the dielectric constant and the resistivity are configured to cause convergence to solutions with fewer than 150 iterations.

20. The one or more non-transitory machine-readable media of claim 18, wherein the instructions to use contraction mapping to iteratively solve for the dielectric constant and the resistivity are configured to cause convergence regardless of initial guesses between 0.001 and 1000 for either resistivity in units of ohm-meter or dielectric constant.