Digital pressure derivative method and program storage device

• US 7,107,188 B2
• Filed: 04/10/2003
• Issued: 09/12/2006
• Est. Priority Date: 01/08/2003
• Status: Expired due to Fees

• Current Assignee: Schlumberger Technology Corporation (Schlumberger NV)
• Original Assignee: Schlumberger Technology Corporation (Schlumberger NV)
• Inventors: Veneruso, Anthony, Spath, Jeff
• Primary Examiner(s): Bui, Bryan
• Assistant Examiner(s): Bhat, Aditya S.
• Application Number:
  US10/411,579
• Publication Number:
  US 20040133350A1
• Time in Patent Office: 1,251 Days
• Field of Search: 702/17, 702/187, 702/195, 340853-8553, 367/82, 375/143
• US Class Current: 702/187
• CPC Class Codes: E21B 49/008 by injection test; by analysing pressure variations in an injection or production test, e.g. for estimating the skin factor measuring pressure E21B47/06
• CPC Class Codes: G01V 1/36 Effecting static or dynamic corrections on records, e.g. correcting spread; Correlating seismic signals; Eliminating effects of unwanted energy
• CPC Class Codes: G01V 2210/32 Noise reduction
• CPC Class Codes: G06F 17/10 Complex mathematical operations function generation by table look-up G06F1/03; evaluation of elementary functions by calculation G06F7/544
• CPC Class Codes: H03H 17/06 Non-recursive filters

Abstract

In a Digital Pressure Derivative Technique (DPDT) which uses convolution, an input signal including noise is convolved with a wavelet to produce an output signal which is a derivative of the input signal and which is substantially devoid of the noise. The DPDT can be used in many applications; however, in one such application, a pressure signal from a wellbore including noise is convolved with a special wavelet to produce an output signal which is the derivative of the input pressure signal, the output signal being substantially devoid of the noise. The wavelet is made from the unit sample response of a bandlimited, optimum linear phase Finite Impulse Response digital differentiator. The DPDT technique enables analysts to customize the differentiation based on the quality of the measured data without having to use traditional, subjective smoothing algorithms. The results presented here, in connection with the one such application mentioned above, bring significant improvements to the diagnosis and interpretation of pressure transient data, and interpretation techniques that utilize higher-order derivatives are now practical.

Claims

1. A method responsive to an input signal representing a measurement and having noise for generating an output signal representing said measurement substantially devoid of said noise, said method comprising:

convolving said input signal, representing said measurement and having noise, with a wavelet having a determined set of coefficients; and

generating said output signal in response to the convolving step, said output signal being substantially devoid of the noise.

2. The method of claim 1, wherein said input signal is a particular function representing a pressure measurement in a wellbore, said pressure measurement including noise.

3. The method of claim 1, wherein said output signal is a derivative of said input signal, said output signal being substantially devoid of said noise.

4. method of claim 3, wherein the step of convolving said input signal with a wavelet includes convolving said input signal with said wavelet using the following equation:

$p_{\mathrm{wf}}<br><br>\left(t\right)={\int <br><br>}_0^t<br><br>q<br><br>\left(\mathrm{\tau <br><br>}\right)<br><br>p_{\mathrm{wf}}^{\mathrm{\prime <br><br>}}<br><br>\left(t-\mathrm{\tau <br><br>}\right)<br><br><br><br>d<br><br>\mathrm{\tau <br><br>}.$

5. The method of claim 4, wherein said wavelet is a derivative wavelet.

6. The method of claim 5, wherein said derivative wavelet is made from a unit sample response of a bandlimited, optimum linear phase Finite Impulse Response digital differentiator.

7. The method of claim 3, wherein the step of convolving said input signal with a wavelet includes convolving said input signal with said wavelet using the following equation:

$L^{-1}<br><br>\left\{f<br><br>\left(s\right)<br><br>g<br><br>\left(s\right)\right\}={\int <br><br>}_0^t<br><br>F<br><br>\left(\mathrm{\tau <br><br>}\right)<br><br>G<br><br>\left(t-\mathrm{\tau <br><br>}\right)<br><br><br><br>d<br><br>\mathrm{\tau <br><br>}.$

8. The method of claim 7, wherein said wavelet is a derivative wavelet.

9. The method of claim 8, wherein said derivative wavelet is made from a unit sample response of a bandlimited, optimum linear phase Finite Impulse Response digital differentiator.

10. The method of claim 1, further comprising:

further convolving said output signal with said wavelet; and

generating a further output signal in response to the further convolving step, said further output signal being substantially devoid of said noise.

11. The method of claim 10, wherein said output signal represents a first derivative of said input signal, said output signal being substantially devoid of said noise.

12. The method of claim 11, wherein said further output signal represents a second derivative of said input signal, said further output signal being substantially devoid of said noise.

13. The method of claim 12, wherein the step of convolving said input signal with a wavelet and the step of further convolving said output signal with said wavelet uses the following equation:

$p_{\mathrm{wf}}<br><br>\left(t\right)={\int <br><br>}_0^t<br><br>q<br><br>\left(\mathrm{\tau <br><br>}\right)<br><br>p_{\mathrm{wf}}^{\mathrm{\prime <br><br>}}<br><br>\left(t-\mathrm{\tau <br><br>}\right)<br><br><br><br>d<br><br>\mathrm{\tau <br><br>}.$

14. The method of claim 13, wherein said wavelet is made from a unit sample response of a bandlimited, optimum linear phase Finite Impulse Response digital differentiator.

15. The method of claim 12, wherein the step of convolving said input signal with a wavelet and the step of further convolving said output signal with said wavelet uses the following equation:

$L^{-1}<br><br>\left\{f<br><br>\left(s\right)<br><br>g<br><br>\left(s\right)\right\}={\int <br><br>}_0^t<br><br>F<br><br>\left(\mathrm{\tau <br><br>}\right)<br><br>G<br><br>\left(t-\mathrm{\tau <br><br>}\right)<br><br><br><br>d<br><br>\mathrm{\tau <br><br>}.$

16. The method of claim 15, wherein said wavelet is made from a unit sample response of a bandlimited, optimum linear phase Finite Impulse Response digital differentiator.

33. The method of claim 1, wherein said input signal is a particular function including a set of digital data samples representing a pressure measurement in a wellbore, said pressure measurement including noise.

34. The method of claim 33, wherein said output signal is a derivative of said particular function, said derivative of said particular function being substantially devoid of said noise.

35. The method of claim 34, wherein the step of convolving said digital input signal with a wavelet includes convolving said input signal with said wavelet using the following equation:

$\mathrm{DPDT}<br><br>\left(t\right)=\underset{m=0}{\overset{N-1}{\sum <br><br>}}<br><br>h<br><br>\left(m\right)<br><br>\mathrm{signal}<br><br>\left(t-m\right).$

36. The method of claim 35, wherein said wavelet is a derivative wavelet.

37. The method of claim 36, wherein said derivative wavelet is made from a band limited transform of an ideal differentiator, the band limited transform having a band limited cutoff frequency design value.

38. The method of claim 37, wherein the band limited cutoff frequency design value is selected as that value above which noise is a principal component of the total signal.

39. The method of claim 36, wherein one or more coefficients of the wavelet are computed using a Chebyshev method using a Remez exchange algorithm.

17. A program storage device readable by a machine, tangibly embodying a program of instructions executable by the machine, to perform method steps for generating an output signal representing a measurement which is substantially devoid of any noise in response to an input signal representing said measurement and which includes said noise, said method steps comprising:

convolving said input signal, representing said measurement and including said noise, with a wavelet having a determined set of coefficients; and

generating said output signal in response to the convolving step, said output signal being substantially devoid of the noise.

18. The program storage device of claim 17, wherein said input signal is a particular function representing a pressure measurement in a wellbore, said pressure measurement including noise.

19. The program storage device of claim 17, wherein said output signal is a derivative of said input signal, said output signal being substantially devoid of said noise.

20. The program storage device of claim 19, wherein the step of convolving said input signal with a wavelet uses the following equation:

$p_{\mathrm{wf}}<br><br>\left(t\right)={\int <br><br>}_0^t<br><br>q<br><br>\left(\mathrm{\tau <br><br>}\right)<br><br>p_{\mathrm{wf}}^{\mathrm{\prime <br><br>}}<br><br>\left(t-\mathrm{\tau <br><br>}\right)<br><br><br><br>d<br><br>\mathrm{\tau <br><br>}.$

21. The program storage device of claim 20, wherein said wavelet is a derivative wavelet.

22. The program storage device of claim 21, wherein said derivative wavelet is made from a unit sample response of a bandlimited, optimum linear phase Finite Impulse Response digital differentiator.

23. The program storage device of claim 19, wherein the step of convolving said input signal with a wavelet uses the following equation:

$L^{-1}<br><br>\left\{f<br><br>\left(s\right)<br><br>g<br><br>\left(s\right)\right\}={\int <br><br>}_0^t<br><br>F<br><br>\left(\mathrm{\tau <br><br>}\right)<br><br>G<br><br>\left(t-\mathrm{\tau <br><br>}\right)<br><br><br><br>d<br><br>\mathrm{\tau <br><br>}.$

24. The program storage device of claim 23, wherein said wavelet is a derivative wavelet.

25. The program storage device of claim 24, wherein said derivative wavelet is made from a unit sample response of a bandlimited, optimum linear phase Finite Impulse Response digital differentiator.

26. The program storage device of claim 17, further comprising:

further convolving said output signal with said wavelet; and

generating a further output signal in response to the further convolving step, said further output signal being substantially devoid of said noise.

27. The program storage device of claim 26, wherein said output signal represents a first derivative of said input signal, said output signal being substantially devoid of said noise.

28. The program storage device of claim 27, wherein said further output signal represents a second derivative of said input signal, said further output signal being substantially devoid of said noise.

29. The program storage device of claim 28, wherein the step of convolving said input signal with a wavelet and the step of further convolving said output signal with said wavelet each use the following equation:

$p_{\mathrm{wf}}<br><br>\left(t\right)={\int <br><br>}_0^t<br><br>q<br><br>\left(\mathrm{\tau <br><br>}\right)<br><br>p_{\mathrm{wf}}^{\mathrm{\prime <br><br>}}<br><br>\left(t-\mathrm{\tau <br><br>}\right)<br><br><br><br>d<br><br>\mathrm{\tau <br><br>}.$

30. The program storage device of claim 29, wherein said wavelet is made from a unit sample response of a bandlimited, optimum linear phase Finite Impulse Response digital differentiator.

31. The storage program device of claim 28, wherein the step of convolving said input signal with a wavelet and the step of further convolving said output signal with said wavelet each use the following equation:

$L^{-1}<br><br>\left\{f<br><br>\left(s\right)<br><br>g<br><br>\left(s\right)\right\}={\int <br><br>}_0^t<br><br>F<br><br>\left(\mathrm{\tau <br><br>}\right)<br><br>G<br><br>\left(t-\mathrm{\tau <br><br>}\right)<br><br><br><br>d<br><br>\mathrm{\tau <br><br>}.$

32. The program storage device of claim 31, wherein said wavelet is made from a unit sample response of a bandlimited, optimum linear phase Finite Impulse Response digital differentiator.

40. The program storage device of claim 17, wherein said input signal is a particular function including a set of digital data samples representing a pressure measurement in a wellbore, said pressure measurement including noise.

41. The program storage device of claim 40, wherein said output signal is a derivative of said particular function, said derivative of said particular function being substantially devoid of said noise.

42. The program storage device of claim 41, wherein the step of convolving said digital input signal with a wavelet includes convolving said input signal with said wavelet using the following equation:

$\mathrm{DPDT}<br><br>\left(t\right)=\underset{m=0}{\overset{N-1}{\sum <br><br>}}<br><br>h<br><br>\left(m\right)<br><br>\mathrm{signal}<br><br>\left(t-m\right).$

43. The program storage device of claim 42, wherein said wavelet is a derivative wavelet.

44. The program storage device of claim 43, wherein said derivative wavelet is made from a band limited transform of the ideal differentiator, said band limited transform having a band limited cutoff frequency design value.

45. The program storage device of claim 44, wherein the band limited cutoff frequency'"'"'s design value is selected as that value above which noise is a principal component of the total signal.

46. The program storage device of claim 43, wherein one or more coefficients of the wavelet are computed using a Chebyshev method using a Remez exchange algorithm.

47. A method responsive to an input signal representing a measurement and having noise for generating an output signal representing said measurement substantially devoid of said noise, said method comprising:

convolving said input signal, representing said measurement and including said noise, with a wavelet having a determined set of coefficients; and

generating said output signal in response to the convolving step,

said output signal representing a derivative of said input signal, said output signal being substantially devoid of said noise.

48. A program storage device readable by a machine, tangibly embodying a program of instructions executable by the machine, to perform method steps which are responsive to an input signal representing a measurement and having noise for generating an output signal representing said measurement substantially devoid of said noise, said method steps comprising:

convolving said input signal, representing a measurement and including said noise, with a wavelet having a determined set of coefficients; and

generating said output signal in response to the convolving step,

said output signal representing a derivative of said input signal, said output signal being substantially devoid of said noise.