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Pure and Applied Geophysics

, Volume 176, Issue 1, pp 315–333 | Cite as

Accurate Jacobian Matrix Using the Exact Zoeppritz Equations and Effects on the Inversion of Reservoir Properties in Porous Media

  • Xiaobo Liu
  • Jingyi ChenEmail author
  • Fuping Liu
  • Anling Wang
  • Zhencong Zhao
Article
  • 88 Downloads

Abstract

The analysis of amplitude variation with offset (AVO) plays a significant role in fluid detection and lithology discrimination in hydrocarbon reservoirs. The Zoeppritz equations are part of the basic theory of AVO analysis which describes the relationship between seismic reflection and transmission coefficients and elastic rock properties (e.g., P- and S-wave velocities and density). Currently, most AVO inversion methods are based on approximations of the exact Zoeppritz equations, which not only limit the accuracy of AVO inversion, but also restrict its application to wide-angle seismic reflection data. In addition, the most difficult part of linear AVO inversion obtaining an accurate Jacobian matrix (partial derivatives of reflection coefficients with respect to inverted parameters). Based on our previous study on the accurate gradient calculation of seismic reflection coefficients for the inversion of rock properties, we further combine the exact Zoeppritz equations with Biot–Gassmann equations to compute the gradients of seismic reflection coefficients with solid density and reservoir properties (e.g., porosity, water/gas/oil saturations) in porous media. In this paper, the partial derivative expressions of the Zoeppritz matrix elements with respect to solid density and reservoir properties are simplified to simple algebraic equations, which are readily calculated. By comparing reflection coefficients and partial derivative curves with those obtained by classic Shuey and Aki–Richards approximations, we show that our proposed method can be used to accurately obtain reservoir properties in AVO inversion.

Keywords

Exact Zoeppritz equations Biot–Gassmann equations AVO inversion accurate solution 

Notes

Acknowledgements

The authors acknowledge the Faculty Internationalization Grant at the University of Tulsa. This work was supported by BIGC Project (Ec201803, Ea201806 and Ed201802), Joint Funding Project of Beijing Municipal Commission of Education Science and Beijing Natural Science Funding Committee (KZ201710015010, KZ201510015015 and PXM2016_014223_000025). This work was also supported by the College Student Research Program of 2016.

References

  1. Adam, L., Batzle, M., & Brevik, I. (2006). Gassmann’s fluid substitution and shear modulus variability in carbonates at laboratory seismic and ultrasonic frequencies. Geophysics, 71, 173–183.CrossRefGoogle Scholar
  2. Aki, K., & Richards, P. (1980). Quantitative seismology—theory and method. New York: W. H. Freeman & Co.Google Scholar
  3. Aki, K., & Richards, P. G. (2002). Quantitative seismology. Sausalito: University Science Books.Google Scholar
  4. Batzle, M., & Wang, Z. (1992). Seismic properties of pore fluids. Geophysics, 57, 1396–1408.CrossRefGoogle Scholar
  5. Biot, M. A. (1941). General theory of three-dimensional consolidation. Journal of Applied Physics, 12, 155–164.CrossRefGoogle Scholar
  6. Biot, M. A. (1962a). Mechanics of deformation and acoustic propagation in porous media. Journal of Applied Physics, 33, 1482–1498.CrossRefGoogle Scholar
  7. Biot, M. A. (1962b). Generalized theory of acoustic propagation in porous dissipative media. Journal of the Acoustical Society of America, 34, 1254–1264.CrossRefGoogle Scholar
  8. Bortfeld, R. (1961). Approximations to the reflection and transmission coefficients of plane longitudinal and transverse waves. Geophysical Prospecting, 9, 485–502.CrossRefGoogle Scholar
  9. Buland, A., & Omre, H. (2003). Bayesian linearized AVO inversion. Geophysics, 68, 185–198.CrossRefGoogle Scholar
  10. Foster, D. J., Keys, R. G., & Lane, F. D. (2010). Interpretation of AVO anomalies. Geophysics, 75, 75A3–75A13.CrossRefGoogle Scholar
  11. Gassmann, F. (1951). Elastic wave through a packing of spheres. Geophysics, 16, 673–685.CrossRefGoogle Scholar
  12. Goodway, B., Chen, T., & Downton, J. (1997). Improved AVO fluid detection and lithology discrimination using Lame petrophysical parameters; “λλ,” “μμ,” and “λ/μ fluid stack,” from P and S inversions. In 67th SEG annual meeting. Technical program expanded abstracts, Dallas, USA (pp. 183–186).Google Scholar
  13. Grana, D., Mukerji, T., Dvorkin, J., & Mavko, G. (2012). Stochastic inversion of facies from seismic data based on sequential simulations and probability perturbation method. Geophysics, 77, M53–M72.CrossRefGoogle Scholar
  14. Hilterman, F. J. (2001) Seismic amplitude interpretation. Distinguished instructor short course. Distinguished instructor series No. 4. Tulsa, USA: Society of Exploration Geophysics (SEG) and European Association of Geoscientists & Engineers (EAGE).Google Scholar
  15. Keys, R. G., & Xu, S. (2002). An approximation for the Xu–White velocity model. Geophysics, 67, 1406–1414.CrossRefGoogle Scholar
  16. Larsen, J. A. (1999). AVO inversion by simultaneous P-P and P-S inversion. Calgary: M. S. Thesis, University of Calgary.Google Scholar
  17. Lehochi, I., Avseth, P., & Hadziavidic, V. (2015). Probabilistic estimation of density and shear information from Zeoppritz’s equation. The Leading Edge, 34, 1036–1047.CrossRefGoogle Scholar
  18. Liu, X., Liu, F., Meng, X., & Xiao, J. (2012a). An accurate method of computing the gradient of seismic wave reflection coefficients (SWRCs) for the inversion of stratum parameters. Surveys In Geophysics, 33, 293–309.CrossRefGoogle Scholar
  19. Liu, F., Meng, X., Wang, Y., Shen, Q., & Yang, C. (2011). Jacobian matrix for the inversion of P- and S-wave velocities and its accurate computation method. Science China Earth Sciences, 54, 647–654.CrossRefGoogle Scholar
  20. Liu, F., Meng, X., Xiao, J., Wang, Y., & Shen, G. (2012b). Applying accurate gradients of seismic wave reflection coefficients (SWRC) to the inversion of seismic wave velocities. Science China Earth Sciences, 55, 1953–1960.CrossRefGoogle Scholar
  21. Lu, J., Wang, Y., Chen, J., & An, Y. (2018). Joint anisotropic amplitude variation with offset inversion of PP and PS seismic data. Geophysics, 83, N31–N50.CrossRefGoogle Scholar
  22. Lu, J., Yang, Z., Wang, Y., & Shi, Y. (2015). Joint PP and PS AVA seismic inversion using exact Zoeppritz equations. Geophysics, 80, 239–250.CrossRefGoogle Scholar
  23. Mavko, G., Chan, C., & Mukerji, T. (1995). Fluid substitution: Estimating changes in Vp without knowing Vs. Geophysics, 60, 1750–1755.CrossRefGoogle Scholar
  24. Ostrander, W. J. (1984). Plane-wave reflection coefficients for gas and sands at non-normal angles of incidence. Geophysics, 49, 1637–1648.CrossRefGoogle Scholar
  25. Reilly, J. M. (1994). Wireline shear and AVO modeling: Application to AVO investigations of the Tertiary, UK. Central North Sea. Geophysics, 59, 1249–1260.CrossRefGoogle Scholar
  26. Richards, P. G., & Frasier, C. W. (1976). Scattering of elastic waves from depth-dependent inhomogeneities. Geophysics, 41, 441–458.CrossRefGoogle Scholar
  27. Russell, B. H., Gray, D., & Hampson, D. P. (2011). Linearized AVO and poroelasticity. Geophysics, 76, C19–C29.CrossRefGoogle Scholar
  28. Russell, B., Hedlin, K., Hilterman, F., & Lines, L. (2003). Fluid-property discrimination with AVO: A Biot–Gassmann perspective. Geophysics, 68, 29–39.CrossRefGoogle Scholar
  29. Shou, H., Liu, H., & Gao, J. H. (2006). AVO inversion based on common shot migration. Applied Geophysics, 3, 99–104.CrossRefGoogle Scholar
  30. Shuey, R. T. (1985). A simplification of the Zoeppritz equations. Geophysics, 50, 609–614.CrossRefGoogle Scholar
  31. Skopintseva, L., Ayzenberg, M., Landro, M., Nefedkina, T., & Aizenberg, A. M. (2011). Long-offset AVO inversion of PP reflections from plane interfaces using effective reflection coefficients. Geophysics, 76, C65–C97.CrossRefGoogle Scholar
  32. Tigrek, S., Slob, E. C., Dillen, M. W. P., Cloetingh, S. A. P. L., & Fokkema, J. T. (2005). Linking dynamic elastic parameters to static state of stress: Toward an integrated approach to subsurface stress analysis. Tectonophysics, 397, 167–179.CrossRefGoogle Scholar
  33. Ursin, B., & Dahl, T. (1992). Seismic reflection amplitudes. Geophysical Prospecting, 40, 483–512.CrossRefGoogle Scholar
  34. Wang, Y. (1999). Approximations to Zoeppritz equations and their use in AVO analysis. Geophysics, 64, 1920–1927.CrossRefGoogle Scholar
  35. White, L., & Castagna, J. P. (2002). Stochastic fluid modulus inversion. Geophysics, 67, 1835–1843.CrossRefGoogle Scholar
  36. Wollner, U., & Dvorkin, J. (2016). Effective fluid and grain bulk moduli for heterogeneous thinly layered poroelastic media. Geophysics, 81, D573–D584.CrossRefGoogle Scholar
  37. Yin, X., & Zhang, S. (2014). Bayesian inversion for effective pore-fluid bulk modulus based on fluid-matrix decoupled amplitude variation with offset approximation. Geophysics, 79, R221–R232.CrossRefGoogle Scholar
  38. Zhi, L., Chen, S., & Li, X. (2016). Amplitude variation with angle inversion using the exact Zoeppritz equations—Theory and methodology. Geophysics, 81, N1–N15.CrossRefGoogle Scholar
  39. Zhu, X., & McMechan, G. (2012). AVO inversion using the Zoeppritz equation for PP reflections. In 82nd SEG annual international meeting. Technical program expanded abstracts, Las Vegas, USA (pp. 1–5).Google Scholar
  40. Zoeppritz, K. (1919). Erdbebenwellen VIII B, Uber die reflexion und durchgang seismischer wellen durch unstetigkeitsflachen. Gottinger Nachr, 1, 66–84.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Xiaobo Liu
    • 1
  • Jingyi Chen
    • 1
    Email author
  • Fuping Liu
    • 2
  • Anling Wang
    • 2
  • Zhencong Zhao
    • 1
  1. 1.Seismic Anisotropy Group, Department of GeosciencesThe University of TulsaTulsaUSA
  2. 2.Beijing Institute of Graphic CommunicationBeijingChina

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