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

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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.

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References

  • 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.

    Article  Google Scholar 

  • Aki, K., & Richards, P. (1980). Quantitative seismology—theory and method. New York: W. H. Freeman & Co.

    Google Scholar 

  • Aki, K., & Richards, P. G. (2002). Quantitative seismology. Sausalito: University Science Books.

    Google Scholar 

  • Batzle, M., & Wang, Z. (1992). Seismic properties of pore fluids. Geophysics, 57, 1396–1408.

    Article  Google Scholar 

  • Biot, M. A. (1941). General theory of three-dimensional consolidation. Journal of Applied Physics, 12, 155–164.

    Article  Google Scholar 

  • Biot, M. A. (1962a). Mechanics of deformation and acoustic propagation in porous media. Journal of Applied Physics, 33, 1482–1498.

    Article  Google Scholar 

  • Biot, M. A. (1962b). Generalized theory of acoustic propagation in porous dissipative media. Journal of the Acoustical Society of America, 34, 1254–1264.

    Article  Google Scholar 

  • Bortfeld, R. (1961). Approximations to the reflection and transmission coefficients of plane longitudinal and transverse waves. Geophysical Prospecting, 9, 485–502.

    Article  Google Scholar 

  • Buland, A., & Omre, H. (2003). Bayesian linearized AVO inversion. Geophysics, 68, 185–198.

    Article  Google Scholar 

  • Foster, D. J., Keys, R. G., & Lane, F. D. (2010). Interpretation of AVO anomalies. Geophysics, 75, 75A3–75A13.

    Article  Google Scholar 

  • Gassmann, F. (1951). Elastic wave through a packing of spheres. Geophysics, 16, 673–685.

    Article  Google Scholar 

  • 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).

  • 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.

    Article  Google Scholar 

  • 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).

  • Keys, R. G., & Xu, S. (2002). An approximation for the Xu–White velocity model. Geophysics, 67, 1406–1414.

    Article  Google Scholar 

  • Larsen, J. A. (1999). AVO inversion by simultaneous P-P and P-S inversion. Calgary: M. S. Thesis, University of Calgary.

    Google Scholar 

  • Lehochi, I., Avseth, P., & Hadziavidic, V. (2015). Probabilistic estimation of density and shear information from Zeoppritz’s equation. The Leading Edge, 34, 1036–1047.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Lu, J., Yang, Z., Wang, Y., & Shi, Y. (2015). Joint PP and PS AVA seismic inversion using exact Zoeppritz equations. Geophysics, 80, 239–250.

    Article  Google Scholar 

  • Mavko, G., Chan, C., & Mukerji, T. (1995). Fluid substitution: Estimating changes in Vp without knowing Vs. Geophysics, 60, 1750–1755.

    Article  Google Scholar 

  • Ostrander, W. J. (1984). Plane-wave reflection coefficients for gas and sands at non-normal angles of incidence. Geophysics, 49, 1637–1648.

    Article  Google Scholar 

  • Reilly, J. M. (1994). Wireline shear and AVO modeling: Application to AVO investigations of the Tertiary, UK. Central North Sea. Geophysics, 59, 1249–1260.

    Article  Google Scholar 

  • Richards, P. G., & Frasier, C. W. (1976). Scattering of elastic waves from depth-dependent inhomogeneities. Geophysics, 41, 441–458.

    Article  Google Scholar 

  • Russell, B. H., Gray, D., & Hampson, D. P. (2011). Linearized AVO and poroelasticity. Geophysics, 76, C19–C29.

    Article  Google Scholar 

  • Russell, B., Hedlin, K., Hilterman, F., & Lines, L. (2003). Fluid-property discrimination with AVO: A Biot–Gassmann perspective. Geophysics, 68, 29–39.

    Article  Google Scholar 

  • Shou, H., Liu, H., & Gao, J. H. (2006). AVO inversion based on common shot migration. Applied Geophysics, 3, 99–104.

    Article  Google Scholar 

  • Shuey, R. T. (1985). A simplification of the Zoeppritz equations. Geophysics, 50, 609–614.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Ursin, B., & Dahl, T. (1992). Seismic reflection amplitudes. Geophysical Prospecting, 40, 483–512.

    Article  Google Scholar 

  • Wang, Y. (1999). Approximations to Zoeppritz equations and their use in AVO analysis. Geophysics, 64, 1920–1927.

    Article  Google Scholar 

  • White, L., & Castagna, J. P. (2002). Stochastic fluid modulus inversion. Geophysics, 67, 1835–1843.

    Article  Google Scholar 

  • Wollner, U., & Dvorkin, J. (2016). Effective fluid and grain bulk moduli for heterogeneous thinly layered poroelastic media. Geophysics, 81, D573–D584.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Zhi, L., Chen, S., & Li, X. (2016). Amplitude variation with angle inversion using the exact Zoeppritz equations—Theory and methodology. Geophysics, 81, N1–N15.

    Article  Google Scholar 

  • 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).

  • Zoeppritz, K. (1919). Erdbebenwellen VIII B, Uber die reflexion und durchgang seismischer wellen durch unstetigkeitsflachen. Gottinger Nachr, 1, 66–84.

    Google Scholar 

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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.

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Authors and Affiliations

  1. Seismic Anisotropy Group, Department of Geosciences, The University of Tulsa, 800 South Tucker Drive, Tulsa, OK, 74104-9700, USA

    Xiaobo Liu, Jingyi Chen & Zhencong Zhao

  2. Beijing Institute of Graphic Communication, Beijing, 102600, China

    Fuping Liu & Anling Wang

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  1. Xiaobo Liu
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  2. Jingyi Chen
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  3. Fuping Liu
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  4. Anling Wang
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  5. Zhencong Zhao
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Correspondence to Jingyi Chen.

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Liu, X., Chen, J., Liu, F. et al. Accurate Jacobian Matrix Using the Exact Zoeppritz Equations and Effects on the Inversion of Reservoir Properties in Porous Media. Pure Appl. Geophys. 176, 315–333 (2019). https://doi.org/10.1007/s00024-018-1969-8

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  • DOI: https://doi.org/10.1007/s00024-018-1969-8

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