Abstract
This paper introduces a new spontaneous potential log model for the case in which formation resistivity is not piecewise constant. The spontaneous potential satisfies an elliptic boundary value problem with jump conditions on the interfaces. It has been shown that the elliptic interface problem has a unique weak solution. Furthermore, a jump condition capturing finite difference scheme is proposed and applied to solve such elliptic problems. Numerical results show validity and effectiveness of the proposed method.
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References
Smits, L. J. M. SP log interpretation in shaly sands. Trans. AIME 243, 123–136 (1968)
Howard, A. Q. A new invasion model for resistivity log interpretation. The Log Analyst 33(2), 96–110 (1992)
Oppenheim, A. V. and Schafer, R. W. Digital Signal Processing, Prentice Hall, New Jersey (1975)
Li, T. T., Tan, Y. J., Peng, Y. J., and Li, H. L. Mathematical methods for the SP well-logging. Applied and Industrial Mathematics (ed. Spigler, R.), Kluwer Academic Publishers, Dordrecht, 343–349 (1991)
Li, T. T., Tan, Y. J., and Peng, Y. J. Mathematical model and method for spontaneous potential well-logging. Eur. J. Appl. Math. 5(2), 123–139 (1994)
Li, T. T. A class of nonlocal boundary value problems for partial differential equations and its applications in numerical analysis. J. Comput. Appl. Math. 28, 49–62 (1989)
Cai, Z. J. Asymptotic behavior for a class of elliptic equivalued surface boundary value problem with discontinuous interface conditions. Appl. Math. J. Chinese Univ. Ser. B 10(3), 237–250 (1995)
Zhang, G. J. Electrical Well Logging (in Chinese), Petroleum Industry Press, Beijing (1984)
Adams, R. A. Sobolev Spaces, Academic Press, New York (1975)
Peng, Y. J. A necessary and sufficient condition for the well-posedness of a class of boundary value problem (in Chinese). Journal of Tongji University 16(1), 91–100 (1988)
Zhou, Y. and Cai, Z. J. Convergence of a numerical method in mathematical spontaneous potential well-logging. Eur. J. Appl. Math. 7(1), 31–41 (1996)
Chew, W. C., Nie, Z. P., and Liu, Q. H. An efficient solution for the response of electrical well logging tools in a complex environment. IEEE Trans. Geosci. Remote Sensing 29(2), 308–313 (1991)
Yuan, N., Nie, X. C., and Nie, Z. P. Numerical analysis of SP log response in complex heterogeneous media (in Chinese). Chinese J. Geophys. 41(suppl.), 429–436 (1998)
Li, H. L. “Exact” solution of SP well-logging equation in homogeneous formation (in Chinese). Chin. Ann. Math. Ser. A. 17(1), 87–96 (1996)
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(Communicated by Zhe-wei ZHOU)
Project supported by the National Natural Science Foundation of China (No. 10431030) and the Shanghai Natural Science Foundation (No. 08ZR1401100)
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Pan, Kj., Tan, Yj. & Hu, Hl. Mathematical model and numerical method for spontaneous potential log in heterogeneous formations. Appl. Math. Mech.-Engl. Ed. 30, 209–219 (2009). https://doi.org/10.1007/s10483-009-0208-z
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DOI: https://doi.org/10.1007/s10483-009-0208-z