SIMULTANEOUS INVERSION OF INDUCTION DATA FOR DIELECTRIC PERMITTIVITY AND ELECTRIC CONDUCTIVITY

• US 20090248308A1
• Filed: 03/16/2009
• Published: 10/01/2009
• Est. Priority Date: 03/28/2008
• Status: Active Grant

• Current Assignee: Schlumberger Technology Corporation (SLB)
• Original Assignee: Schlumberger Technology Corporation (SLB)
• Inventors: Luling, Martin
• Granted Patent: US 8,135,542 B2
• Time in Patent Office: Days
• US Class Current: 702/7
• CPC Class Codes: G01V 3/28 using induction coils

Abstract

A method of inverting induction logging data for evaluating the properties of underground formations surrounding a borehole, the data comprising induction voltage measurements obtained from a tool placed close to the formations of interest, the method comprises: (a) defining a relationship relating the induction voltage to wave number, dielectric permittivity and conductivity; defining a cubic polynomial expansion of the relationship; and solving the cubic polynomial relationship using the voltage measurements to obtain values for conductivity that includes skin-effect correction, and apparent dielectric permittivity; and (b) using the obtained values for conductivity and apparent dielectric permittivity to derive a simulated value of induction voltage; determining the difference between the simulated value of the induction voltage and the measured induction voltage; and iteratively updating the values of conductivity and dielectric permittivity used for the derivation of the simulated value of induction voltage to minimise its difference with respect to the measured value.

Claims

1. A method of inverting induction logging data for evaluating the properties of underground formations surrounding a borehole, the data comprising induction voltage measurements obtained from a tool placed close to the formations of interest, the method comprising:

(a) defining a relationship relating the induction voltage to wave number, dielectric permittivity and conductivity;

defining a cubic polynomial expansion of the relationship; and

solving the cubic polynomial relationship using the voltage measurements to obtain values for conductivity that includes skin-effect correction, and apparent dielectric permittivity; and

(b) using the obtained values for conductivity and apparent dielectric permittivity to derive a simulated value of induction voltage;

determining the difference between the simulated value of the induction voltage and the measured induction voltage; and

iteratively updating the values of conductivity and dielectric permittivity used for the derivation of the simulated value of induction voltage to minimise its difference with respect to the measured value.

2. A method as claimed in claim 1, wherein the relationship relating induction voltage to wave number, dielectric permittivity and conductivity is:

$k=\frac{\mathrm{\omega <br><br>}}{c}<br><br>\sqrt{{\mathrm{\mu <br><br>}}_r}<br><br>\sqrt{{\mathrm{\varepsilon <br><br>}}_r+\mathrm{<br><br>}<br><br>\frac{\mathrm{\sigma <br><br>}}{{\mathrm{\omega <br><br>\varepsilon <br><br>}}_0}}$

where k is wave number, ω

is circular frequency, c is the speed of light in a vacuum, μ

_r is relative magnetic permeability, ε

_r is relative permittivity, ε

₀ is the absolute dielectric permittivity of a vacuum and σ

is conductivity.

3. A method as claimed in claim 1, comprising deriving the roots of the cubic polynomial and using at least one of the roots to obtain the wave number.

4. A method as claimed in claim 3, comprising ignoring roots giving physically impossible values for parameters of interest.

5. A method as claimed in claim 1, wherein the step of iteratively updating the values of conductivity and dielectric permittivity used for the derivation of the simulated value of induction voltage to minimise its difference with respect to the measured value comprises minimising the squared difference between measured induction voltage U_meas and their simulated reproduction U_simul according to the relationship:

$L=\frac{1}{2}<br><br>{\left(U_{\mathrm{meas}}-U_{\mathrm{simul}}\right)}^2\Rightarrow <br><br>\frac{<br><br>}{<br><br>k}<br><br>L=\left(U_{\mathrm{meas}}-U_{\mathrm{simul}}\right)<br><br>\frac{<br><br>U_{\mathrm{simul}}}{<br><br>k}=0.$

6. A method according to claim 5, wherein the iteration uses a Newton-Raphson algorithm

$U_{\mathrm{meas}}-U_{\mathrm{simul}}^{\left(n-1\right)}=\frac{}{}$

U simul ( n - 1 ) 

k 

( k ( n ) - k ( n - 1 ) )

and the iterative update for the wave number k⁽ⁿ⁾ is derives from the relationship;

$k^{\left(n\right)}=k^{\left(n-1\right)}+\left(U_{\mathrm{meas}}-U_{\mathrm{simul}}^{\left(n-1\right)}\right)/\left(<br><br>U_{\mathrm{simul}}^{\left(n-1\right)}/<br><br>k\right).$

7. A method as claimed in claim 1, wherein the induction voltage measurements comprise in-phase and quadrature measurements of substantially the same accuracy and resolution.

8. A method as claimed in claim 1, wherein the induction logging data is obtained from a three-coil tool comprising a transmitter coil with magnetic dipole moment M_T, a main receiver coil with magnetic dipole moment M₁ at distance r₁ from the transmitter coil, and a bucking coil with a magnetic moment M₂ at distance r₂ from the transmitter coil, the dipole moments being aligned parallel to the axis of the tool, the cubic polynomial expansion comprising:

$U_l=-\mathrm{<br><br>\omega <br><br>\mu <br><br>}<br><br><br><br>\frac{M_T}{2<br><br>\mathrm{\pi <br><br>}}<br><br>\left(\left(\frac{M_1}{2<br><br>r_1}-\frac{M_2}{2<br><br>r_2}\right)<br><br>k^2+\mathrm{<br><br>}<br><br>\left(\frac{M_1}{3}-\frac{M_2}{3}\right)<br><br>k^3\right)$

wherein U_l is the induction voltage.

9. A method as claimed in claim 8, wherein the simulated value of induction voltage is derived according to the relationship

$U_l=-$

ω

μ





M T 2 

π



( M 1 









kr 1 r 1 3 

( 1 - 





kr 1 ) - M 2 









kr 2 r 2 3 

( 1 - 





kr 2 ) )

using a sensitivity function derived according to the relationship;

$\frac{<br><br>U_l}{<br><br>k}=-\mathrm{<br><br>\omega <br><br>\mu <br><br>}<br><br><br><br>\frac{M_T}{2<br><br>\mathrm{\pi <br><br>}}<br><br>\left(\frac{M_1<br><br>{\mathrm{<br><br>}}^{\mathrm{<br><br>}<br><br><br><br>{\mathrm{kr}}_1}}{r_1}-\frac{M_2<br><br>{\mathrm{<br><br>}}^{\mathrm{<br><br>}<br><br><br><br>{\mathrm{kr}}_2}}{r_2}\right)<br><br>k$

where e^ikr is the full electromagnetic wave.

10. A method as claimed in claim 1, wherein the induction logging data is obtained from a three-coil tool comprising a transmitter coil with magnetic dipole moment M_T, a main receiver coil with magnetic dipole moment M₁ at distance r₁ from the transmitter coil, and a bucking coil with a magnetic moment M₂ at distance r₂ from the transmitter coil, the dipole moments being aligned transverse to the axis of the tool, the cubic polynomial expansion comprising:

$U_t=-\mathrm{<br><br>\omega <br><br>\mu <br><br>}<br><br>\frac{<br><br>M_t}{4<br><br>\mathrm{\pi <br><br>}}<br><br>\left(\left(\frac{M_1}{2<br><br>r_1}-\frac{M_2}{2<br><br>r_2}\right)<br><br>k^2+2<br><br>\mathrm{<br><br>}<br><br>\left(\frac{M_1}{3}-\frac{M_2}{3}\right)<br><br>k^3\right)$

wherein U_t is the induction voltage.

11. A method as claimed in claim 10, wherein the simulated value of induction voltage is derived according to the relationship

$U_t=-$

ω

μ





M T 4 

π



( M 1 









kr 1 r 1 3 

( 1 - 





kr 1 - k 2 

r 1 2 ) - M 2 









kr 2 r 2 3 

( 1 - 





kr 2 - k 2 

r 2 2 ) )

using a sensitivity function derived according to the relationship;

$\frac{<br><br>U_t}{<br><br>k}=\mathrm{<br><br>\omega <br><br>\mu <br><br>}<br><br>\frac{<br><br>M_T}{4<br><br>\mathrm{\pi <br><br>}}<br><br>\left(\frac{M_1<br><br>{\mathrm{<br><br>}}^{\mathrm{<br><br>}<br><br><br><br>{\mathrm{kr}}_1}}{r_1}<br><br>\left(1+\mathrm{<br><br>}<br><br><br><br>{\mathrm{kr}}_1\right)-\frac{M_2<br><br>{\mathrm{<br><br>}}^{\mathrm{<br><br>}<br><br><br><br>{\mathrm{kr}}_2}}{r_2}<br><br>\left(1+\mathrm{<br><br>}<br><br><br><br>{\mathrm{kr}}_2\right)\right)<br><br>k$

where e^ikr is the full electromagnetic wave.