New Regularization Algorithms for Solving the Deconvolution Problem in Well Test Data Interpretation

ABSTRACT

Two new regularization algorithms for solving the first-kind Volterra integral equation, which describes the pressure-rate deconvolution problem in well test data interpretation, are developed in this paper. The main features of the problem are the strong nonuniform scale of the solution and large errors (up to 15%) in the input data. In both algorithms, the solution is represented as decomposition on special basic functions, which satisfy given a priori information on solution, and this idea allow us significantly to improve the quality of approximate solution and simplify solving the minimization problem. The theoretical details of the algorithms, as well as the results of numerical experiments for proving robustness of the algorithms, are presented.

Two new regularization algorithms for solving the first-kind Volterra integral equation, which describes the pressure-rate deconvolution problem in well test data interpretation, are developed in this paper. The main features of the problem are the strong nonuniform scale of the solution and large errors (up to 15%) in the input data. In both algorithms, the solution is represented as decomposition on special basic functions, which satisfy given a priori information on solution, and this idea allow us significantly to improve the quality of approximate solution and simplify solving the minimization problem. The theoretical details of the algorithms, as well as the results of numerical experiments for proving robustness of the algorithms, are presented.

KEYWORDS

Deconvolution Problem, Volterra Equations, Well Test, Regularization Algorithm, Quasi-Solutions Method, Tikhonov Regularization, A Priori Information, Discrete Approximation, Non-Quadratic Stabilizing Functional, Special Basis

Deconvolution Problem, Volterra Equations, Well Test, Regularization Algorithm, Quasi-Solutions Method, Tikhonov Regularization, A Priori Information, Discrete Approximation, Non-Quadratic Stabilizing Functional, Special Basis

Cite this paper

nullV. Vasin, G. Skorik, E. Pimonov and F. Kuchuk, "New Regularization Algorithms for Solving the Deconvolution Problem in Well Test Data Interpretation,"*Applied Mathematics*, Vol. 1 No. 5, 2010, pp. 387-399. doi: 10.4236/am.2010.15051.

nullV. Vasin, G. Skorik, E. Pimonov and F. Kuchuk, "New Regularization Algorithms for Solving the Deconvolution Problem in Well Test Data Interpretation,"

References

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[2] D. Bourdet, J. A. Ayoub and Y. M. Pirard, “Use of Pressure Derivative in Well-Test Interpretation,” The Study of Political Economy and Free Enterprise, Vol. 4, No. 2, 1989, pp. 293-302.

[3] K. H. Coats, L. A. Rapoport, J. R. McCord and W. P. Drews, “Determination of Aquifer Influence Functions from Field Data,” Transactions of American Institute of Mining, Metallurgical, and Petroleum Engineers, Vol. 231, 1964, pp. 1417-1424.

[4] F. J. Kuchuk, R. G. Carter and L. Ayestaran, “Decon- volution of Wellbore Pressure and Flow Rate,” The Study of Political Economy and Free Enterprise, Vol. 5, No. 1, 1990, pp. 53-59.

[5] R. J. Gajdica, R. A. Wattenbarger and R. A. Startzman, “A New Method of Matching Aquifer Performance and Determining Original Gas in Place,” The Study of Po- litical Economy and Free Enterprise, Vol. 3, No. 3, 1988, pp. 985-994.

[6] T. S. Hutchinson and V. J. Sikora, “A Generalized Water- Drive Analysis,” Transactions of American Institute of Mining, Metallurgical, and Petroleum Engineers, Vol. 216, 1959, pp. 169-178.

[7] D. L. Katz, M. R. Tek and S. C. Jones, “A Generalized Model for Predicting the Performance of Gas Reservoirs Subject to Water Drive,” SPE Annual Meeting, Los Angeles, 07-10 October 1962.

[8] J. R. Jargon and H. K. van Poollen, “Unit Response Function from Varying-Rate Data,” Journal of Petroleum Technology, Vol. 17, No. 8, August 1965, pp. 965-969.

[9] L. G. Thompson and A. C. Reynolds, “Analysis of Variable-Rate Well-Test Pressure Data Using Duhamel’s Principle,” The Study of Political Economy and Free Enterprise, Vol. 1, No. 5, 1986, pp. 453-469.

[10] J. N. Bostic, R. G. Agarwal and R. D. Carter, “Combined Analysis of Postfracturing Performance and Pressure Buildup Data for Evaluating an MHF Gas Well,” Journal of Petroleum Technology, Vol. 32, No. 10, 1980, pp. 1711-1719.

[11] F. J. Kuchuk, “Applications of Convolution and Deconvolution to Transient Well Tests,” The Study of Political Economy and Free Enterprise, Vol. 5, No. 4, 1990, pp. 375-384.

[12] F. Kucuk and L. Ayestaran, “Analysis of Simultaneously Measured Pressure and Sandface Flow Rate in Transient Well Testing,” Journal of Petroleum Technology, Vol. 37, No. 2, 1985, pp. 323-334.

[13] B. Baygun, F. J. Kuchuk and O. Arikan, “Deconvolution under Normalized Autocorrelation Constraints,” Society of Petroleum Engineers Journal, Vol. 2, No. 3, 1997, pp. 246-253.

[14] A. Roumboutsos and G. Stewart, “A Direct Decon- volution or Convolution Algorithm for Well Test Analysis,” SPE Annual Technical Conference and Exhibition, Houston, Texas, 2-5 October 1988, SPE 18157.

[15] L. C. C. Mendes, M. Tygel and A. C. F. Correa, “A Deconvolution Algorithm for Analysis of Variable-Rate Well Test Pressure Data,” SPE Annual Technical Con- ference and Exhibition, San Antonio, Texas, 8-11 October 1989, SPE 19815.

[16] M. J. Bourgeois and R. N. Horne, “Well-Test-Model Recognition with Laplace Space,” The Study of Political Economy and Free Enterprise, Vol. 8, No. 1, 1993, pp. 17-25.

[17] M. Onur and A. C. Reynolds, “Numerical Laplace Trans- formation of Sampled Data for Well-Test Analysis,” Sustainable Project for Education, Renewable Energy and Environment, Vol. 1, No. 2, 1998, pp. 268-277.

[18] T. von Schroeter, F. Hollaender and A. Gringarten, “Deconvolution of Well Test Data as a Nonlinear Total Least Squares Problem,” SPE Annual Technical Con- ference and Exhibition, New Orleans, Louisiana, 30 September - 3 October 2001, SPE 71574.

[19] T. von Schroeter, F. Hollaender and A. Gringarten, “Analysis of Well Test Data from Permanent Downhole Gauges by Deconvolution,” SPE Annual Technical Conference and Exhibition, San Antonio, 29 September - 2 October 2002.

[20] M. M. Levitan, “Practical Application of Pressure-Rate Deconvolution to Analysis of Real Well Tests,” SPE Annual Technical Conference and Exhibition, Denver, Colorado, 5-8 October 2003.

[21] M. M. Levitan, G. E. Crawford and A. Hardwick, “Practical Considerations for Pressure-Rate Deconvolution of Well Test Data,” SPE Annual Technical Conference and Exhibition, Houston, 26-29 September 2004.

[22] E. A. Pimonov, M. Onur and F. J. Kuchuk, “A New Robust Algorithm for Solution of Pressure-Rate Deconvolution Problem,” Journal of Inverse and Ill-Posed Problems, Vol. 17, No. 6, 2009, pp. 611-627.

[23] E. Pimonov, C. Ayan, M. Onur and F. J. Kuchuk, “A New Pressure-Rate Deconvolution Algorithm to Analyze Wireline Formation Tester and Well-Test Data,” SPE Annual Technical Conference and Exhibition, New Orleans, 4-7 October 2009.

[24] A. L. Ageev, T. V. Antonova, V. V. Vasin, E. A. Pimo- nov and F. J. Kuchuk, “Modified Levenberg-Marquardt Method for Deconvolution Problem Solution,” Youth International Scientific School-Conference “Theory and Numerical Methods for Solution of Inverse and Ill-Posed Problems”, Novosibirsk, 10-20 August 2009, Book of Abstracts, pp. 13-15. (in Russian)

[25] D. Ilk, D. M. Anderson, P. P. Valko and T. A. Bla- singame, “Analysis of Gas-Well Reservoir Performance Data Using - Spline Deconvolution,” SPE Gas Tech- nology Symposium, Calgary, Alberta, 15-17 May 2006.

[26] M. Andrecut and A. M. Madni, “Pressure-Rate Decon- volution Using Non-Orthogonal Exponential Functions Dictionary,” Journal of Integrated Systems, Design, & Process Science, Vol. 11, No. 4, 2007, pp. 41-63.

[27] M. Andrecut, “Pressure Rate Deconvolution Methods for Well Test Analysis,” Modern Physics Letters, Vol. 23, No. 8, 2009, pp. 1027-1051.

[28] M. Cinar, D. Ilk, M. Onur, P. P. Valko and T. A. Blasingame, “A Comparative Study of Recent Robust Deconvolution Algorithms for Well-Test and Production- Data Analysis,” SPE Annual Technical Conference and Exhibition, San Antonio, 24-27 September 2006.

[29] V. V. Vasin, G. G. Skorik, E. A. Pimonov and F. J. Kuchuk, “Regular Methods for Solution of Inverse Problem in Wellbore Geophysics,” 40th All-Russian Youth Conference “Problems of Theoretical and Applied Mathematics”, Ekaterinburg, 26-30 January 2009, pp. 76-82. (in Russian)

[30] V. V. Vasin, G. G. Skorik, E. A. Pimonov and F. J. Kuchuk, “Application of Regular Methods for Solution of the Problem, Arisen at Well Tests Interpretation,” International Conference “Contemporary Problems of Computational Mathematics and Mathematical Physics”, In: Memory of the Academician Alexander A. Samarskii: Proceedings, Moscow, 16-18 June 2009, pp. 33-34. (in Russian)

[31] A. N. Tikhonov and V. Y. Arsenina, “Methods of Solving Ill-Posed Problems,” Nauka English Translation, Moscow; Wiley, New York, 1977.

[32] V. K. Ivanov, V. V. Vasin and V. P. Tanana, “Theory of Linear Ill-Posed Problems and Applications,” Utrecht etc., VSP, 2002; Translation from Russian Edition, 1978.

[33] A. N. Kolmogorov and S. V. Fomin, “Elements of Theory of Functions and Functional Analysis,” Nauka, Moscow, 1981. (in Russian)

[34] A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov and A. G. Yagola, “Regularizing Algorithms and a Priori Information,” Nauka, Moscow, 1983.

[35] V. V. Vasin, “The Method of Quasi-Solutions Ivanov is the Effective Method of Solving Ill-Posed Problems,” Journal of Inverse and Ill-Posed Problems, Vol. 16, No. 6, 2008, pp. 12-24.

[36] F. Stummel, “Discrete Konvergents Linearer Operatoren 1,2,” Mathematische Annalen, Vol. 190, No. 1, 1971, pp. 45-92; Zentralblatt Math, Vol. 120, No. 2, 1971, pp. 230-264.

[37] R. D. Grigorieff, “Zur Theorie Approximations Regularer Operatoren I, II,” Mathematische Nachrichten, Vol. 55, No. 3, 1973, pp. 233-249, 251-263.

[38] G. Vainikko, “Funktionalanalysis der Diskretisierungs- methoden,” Teubner Verlagsgesellschaft, Leipzig, 1976.

[39] V. V. Vasin and A. L. Ageev, “Ill-Posed Problems with a Priori Information,” VSP, Utrecht, 1995.

[40] L. A. Lyusternik and V. I. Sobolev, “Elementy Funk- tcionalnogo Analiza,” Nauka, Moscow, 1965 (in Russian); Translation: “Elements of Functional Analysis,” Ungar, New York, 1965.

[41] F. P. Vasilyev, “Numerical Methods for Solving Extremal Problems,” Nauka, Moscow, 1988. (in Russian)

[42] R. A. Adams, “Sobolev Spaces,” Academic Press, New York, 1975.

[43] G. Eason, B. Noble and I. N. Sneddon, “On Certain Integrals of Lipschitz-Hankel Type Involving Products of Bessel Functions,” The Philosophical Transactions of the Royal Society, London, Vol. A247, April 1955, pp. 529- 551.

[1] A. F. van Everdingen and W. Hurst, “The Application of the Laplace Transformation to Flow Problems in Reservoirs,” Transactions of American Institute of Min- ing, Metallurgical, and Petroleum Engineers, Vol. 186, 1949, pp. 305-324.

[2] D. Bourdet, J. A. Ayoub and Y. M. Pirard, “Use of Pressure Derivative in Well-Test Interpretation,” The Study of Political Economy and Free Enterprise, Vol. 4, No. 2, 1989, pp. 293-302.

[3] K. H. Coats, L. A. Rapoport, J. R. McCord and W. P. Drews, “Determination of Aquifer Influence Functions from Field Data,” Transactions of American Institute of Mining, Metallurgical, and Petroleum Engineers, Vol. 231, 1964, pp. 1417-1424.

[4] F. J. Kuchuk, R. G. Carter and L. Ayestaran, “Decon- volution of Wellbore Pressure and Flow Rate,” The Study of Political Economy and Free Enterprise, Vol. 5, No. 1, 1990, pp. 53-59.

[5] R. J. Gajdica, R. A. Wattenbarger and R. A. Startzman, “A New Method of Matching Aquifer Performance and Determining Original Gas in Place,” The Study of Po- litical Economy and Free Enterprise, Vol. 3, No. 3, 1988, pp. 985-994.

[6] T. S. Hutchinson and V. J. Sikora, “A Generalized Water- Drive Analysis,” Transactions of American Institute of Mining, Metallurgical, and Petroleum Engineers, Vol. 216, 1959, pp. 169-178.

[7] D. L. Katz, M. R. Tek and S. C. Jones, “A Generalized Model for Predicting the Performance of Gas Reservoirs Subject to Water Drive,” SPE Annual Meeting, Los Angeles, 07-10 October 1962.

[8] J. R. Jargon and H. K. van Poollen, “Unit Response Function from Varying-Rate Data,” Journal of Petroleum Technology, Vol. 17, No. 8, August 1965, pp. 965-969.

[9] L. G. Thompson and A. C. Reynolds, “Analysis of Variable-Rate Well-Test Pressure Data Using Duhamel’s Principle,” The Study of Political Economy and Free Enterprise, Vol. 1, No. 5, 1986, pp. 453-469.

[10] J. N. Bostic, R. G. Agarwal and R. D. Carter, “Combined Analysis of Postfracturing Performance and Pressure Buildup Data for Evaluating an MHF Gas Well,” Journal of Petroleum Technology, Vol. 32, No. 10, 1980, pp. 1711-1719.

[11] F. J. Kuchuk, “Applications of Convolution and Deconvolution to Transient Well Tests,” The Study of Political Economy and Free Enterprise, Vol. 5, No. 4, 1990, pp. 375-384.

[12] F. Kucuk and L. Ayestaran, “Analysis of Simultaneously Measured Pressure and Sandface Flow Rate in Transient Well Testing,” Journal of Petroleum Technology, Vol. 37, No. 2, 1985, pp. 323-334.

[13] B. Baygun, F. J. Kuchuk and O. Arikan, “Deconvolution under Normalized Autocorrelation Constraints,” Society of Petroleum Engineers Journal, Vol. 2, No. 3, 1997, pp. 246-253.

[14] A. Roumboutsos and G. Stewart, “A Direct Decon- volution or Convolution Algorithm for Well Test Analysis,” SPE Annual Technical Conference and Exhibition, Houston, Texas, 2-5 October 1988, SPE 18157.

[15] L. C. C. Mendes, M. Tygel and A. C. F. Correa, “A Deconvolution Algorithm for Analysis of Variable-Rate Well Test Pressure Data,” SPE Annual Technical Con- ference and Exhibition, San Antonio, Texas, 8-11 October 1989, SPE 19815.

[16] M. J. Bourgeois and R. N. Horne, “Well-Test-Model Recognition with Laplace Space,” The Study of Political Economy and Free Enterprise, Vol. 8, No. 1, 1993, pp. 17-25.

[17] M. Onur and A. C. Reynolds, “Numerical Laplace Trans- formation of Sampled Data for Well-Test Analysis,” Sustainable Project for Education, Renewable Energy and Environment, Vol. 1, No. 2, 1998, pp. 268-277.

[18] T. von Schroeter, F. Hollaender and A. Gringarten, “Deconvolution of Well Test Data as a Nonlinear Total Least Squares Problem,” SPE Annual Technical Con- ference and Exhibition, New Orleans, Louisiana, 30 September - 3 October 2001, SPE 71574.

[19] T. von Schroeter, F. Hollaender and A. Gringarten, “Analysis of Well Test Data from Permanent Downhole Gauges by Deconvolution,” SPE Annual Technical Conference and Exhibition, San Antonio, 29 September - 2 October 2002.

[20] M. M. Levitan, “Practical Application of Pressure-Rate Deconvolution to Analysis of Real Well Tests,” SPE Annual Technical Conference and Exhibition, Denver, Colorado, 5-8 October 2003.

[21] M. M. Levitan, G. E. Crawford and A. Hardwick, “Practical Considerations for Pressure-Rate Deconvolution of Well Test Data,” SPE Annual Technical Conference and Exhibition, Houston, 26-29 September 2004.

[22] E. A. Pimonov, M. Onur and F. J. Kuchuk, “A New Robust Algorithm for Solution of Pressure-Rate Deconvolution Problem,” Journal of Inverse and Ill-Posed Problems, Vol. 17, No. 6, 2009, pp. 611-627.

[23] E. Pimonov, C. Ayan, M. Onur and F. J. Kuchuk, “A New Pressure-Rate Deconvolution Algorithm to Analyze Wireline Formation Tester and Well-Test Data,” SPE Annual Technical Conference and Exhibition, New Orleans, 4-7 October 2009.

[24] A. L. Ageev, T. V. Antonova, V. V. Vasin, E. A. Pimo- nov and F. J. Kuchuk, “Modified Levenberg-Marquardt Method for Deconvolution Problem Solution,” Youth International Scientific School-Conference “Theory and Numerical Methods for Solution of Inverse and Ill-Posed Problems”, Novosibirsk, 10-20 August 2009, Book of Abstracts, pp. 13-15. (in Russian)

[25] D. Ilk, D. M. Anderson, P. P. Valko and T. A. Bla- singame, “Analysis of Gas-Well Reservoir Performance Data Using - Spline Deconvolution,” SPE Gas Tech- nology Symposium, Calgary, Alberta, 15-17 May 2006.

[26] M. Andrecut and A. M. Madni, “Pressure-Rate Decon- volution Using Non-Orthogonal Exponential Functions Dictionary,” Journal of Integrated Systems, Design, & Process Science, Vol. 11, No. 4, 2007, pp. 41-63.

[27] M. Andrecut, “Pressure Rate Deconvolution Methods for Well Test Analysis,” Modern Physics Letters, Vol. 23, No. 8, 2009, pp. 1027-1051.

[28] M. Cinar, D. Ilk, M. Onur, P. P. Valko and T. A. Blasingame, “A Comparative Study of Recent Robust Deconvolution Algorithms for Well-Test and Production- Data Analysis,” SPE Annual Technical Conference and Exhibition, San Antonio, 24-27 September 2006.

[29] V. V. Vasin, G. G. Skorik, E. A. Pimonov and F. J. Kuchuk, “Regular Methods for Solution of Inverse Problem in Wellbore Geophysics,” 40th All-Russian Youth Conference “Problems of Theoretical and Applied Mathematics”, Ekaterinburg, 26-30 January 2009, pp. 76-82. (in Russian)

[30] V. V. Vasin, G. G. Skorik, E. A. Pimonov and F. J. Kuchuk, “Application of Regular Methods for Solution of the Problem, Arisen at Well Tests Interpretation,” International Conference “Contemporary Problems of Computational Mathematics and Mathematical Physics”, In: Memory of the Academician Alexander A. Samarskii: Proceedings, Moscow, 16-18 June 2009, pp. 33-34. (in Russian)

[31] A. N. Tikhonov and V. Y. Arsenina, “Methods of Solving Ill-Posed Problems,” Nauka English Translation, Moscow; Wiley, New York, 1977.

[32] V. K. Ivanov, V. V. Vasin and V. P. Tanana, “Theory of Linear Ill-Posed Problems and Applications,” Utrecht etc., VSP, 2002; Translation from Russian Edition, 1978.

[33] A. N. Kolmogorov and S. V. Fomin, “Elements of Theory of Functions and Functional Analysis,” Nauka, Moscow, 1981. (in Russian)

[34] A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov and A. G. Yagola, “Regularizing Algorithms and a Priori Information,” Nauka, Moscow, 1983.

[35] V. V. Vasin, “The Method of Quasi-Solutions Ivanov is the Effective Method of Solving Ill-Posed Problems,” Journal of Inverse and Ill-Posed Problems, Vol. 16, No. 6, 2008, pp. 12-24.

[36] F. Stummel, “Discrete Konvergents Linearer Operatoren 1,2,” Mathematische Annalen, Vol. 190, No. 1, 1971, pp. 45-92; Zentralblatt Math, Vol. 120, No. 2, 1971, pp. 230-264.

[37] R. D. Grigorieff, “Zur Theorie Approximations Regularer Operatoren I, II,” Mathematische Nachrichten, Vol. 55, No. 3, 1973, pp. 233-249, 251-263.

[38] G. Vainikko, “Funktionalanalysis der Diskretisierungs- methoden,” Teubner Verlagsgesellschaft, Leipzig, 1976.

[39] V. V. Vasin and A. L. Ageev, “Ill-Posed Problems with a Priori Information,” VSP, Utrecht, 1995.

[40] L. A. Lyusternik and V. I. Sobolev, “Elementy Funk- tcionalnogo Analiza,” Nauka, Moscow, 1965 (in Russian); Translation: “Elements of Functional Analysis,” Ungar, New York, 1965.

[41] F. P. Vasilyev, “Numerical Methods for Solving Extremal Problems,” Nauka, Moscow, 1988. (in Russian)

[42] R. A. Adams, “Sobolev Spaces,” Academic Press, New York, 1975.

[43] G. Eason, B. Noble and I. N. Sneddon, “On Certain Integrals of Lipschitz-Hankel Type Involving Products of Bessel Functions,” The Philosophical Transactions of the Royal Society, London, Vol. A247, April 1955, pp. 529- 551.