Abstract
Petrophysical seismic inversion, aided by rock physics, aims at estimating reservoir properties based on reflection events, but it is generally based on the Gassmann equation, which precludes its applicability to complex reservoirs. To overcome this problem, we present a methodology based on the double-porosity Biot–Rayleigh (BR) model, which takes into account the rock heterogeneities. The volume ratio of inclusions in the BR model is treated as a spatially varying parameter, facilitating a better description of the pore microstructure. The method includes the Zoeppritz equations to extract reservoir properties from prestack data. To handle the ill-posedness of the inversion and achieve a stable solution, the algorithm is formulated as a multi-objective optimization based on the Bayes theorem, where the reservoir-property estimation is jointly conditioned to seismic and elastic data with multiple prior terms. The method is validated with field data of a tight gas sandstone reservoir, illustrating its effectiveness compared to the Gassmann-based estimation, reducing uncertainties and improving the accuracy of identifying gas zones.
Article Highlights
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The petrophysical seismic inversion is based on the double-porosity Biot–Rayleigh model
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Spatially varying inclusion volumes are used to describe complex pore structures
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A multi-objective optimization with joint data misfit enables stable results
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Acknowledgements
We appreciate the editor and two anonymous reviewers for their valuable comments. This work is supported by the National Nature Science Foundation of China (41974123, 42104128), the Jiangsu Province Science Fund for Distinguished Young Scholars (BK20200021), and the Natural Science Foundation of Zhejiang Province (LQ21D040001).
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Appendices
Appendix A: The Biot–Rayleigh equation and its plane-wave solution
Ba et al. (2011) proposed the Biot–Rayleigh model to describe the seismic wave propagation in a double-porosity medium. The governing equations are
where u, U(1), and U(2) are the average particle displacements of the skeleton, and the average fluid displacements in the host and inclusion, respectively, with volume strains ε, ζ(1), and ζ(2), ϕ10 and ϕ20 are the porosities of the host and inclusion with their absolute porosities ϕ1 and ϕ2, κ1 and η are the host-medium permeability and fluid viscosity, respectively, ς denotes the fluid strain increment in the local fluid flow and Ro is the radius of inclusion. The equations contain six stiffness parameters A, N, Q1, Q2, R1, R2, five density coefficients ρ11, ρ12, ρ13, ρ22, ρ33, and two Biot dissipation coefficients b1 and b2.
By substituting a plane-wave kernel into Eqs. (19)–(20), the complex wave number k can be obtained from
where
and
Equation (23) yields three roots, and we choose the fast P-wave one (the classical compressional wave). The phase velocity is given by Carcione (2014) as
where ω is the angular frequency.
Appendix B: Update of posterior weights
Jalobeanu et al. (2002) and Guo et al. (2021a) proposed to update the regularization parameters by using the Monte-Carlo-based maximum likelihood method. We hereby extend the method to be applicable to the multi-objective optimization problem with joint data misfits.
The likelihood function of Eq. (15) is
where the Ω denotes the data space of z. Given the known β, we have
where F1 and F2 denote the seismic and elastic misfit terms in Eq. (12), and
are the normalization constants.
The prior distribution of P(z|β) is
where F3 denotes the prior term in Eq. (12), and
is a normalization constant.
By substituting Eqs. (28) and (30) into (27), the likelihood function of β can be expressed as the negative logarithm of P(dseis,delas|β)
with
By employing the Gauss–Newton descent method to minimize the log-likelihood function 32, β can be iteratively updated as
with
Introducing the expectation of z, regarding its probability distribution, Eq. (35) can be estimated from the expectations of one prior distribution (Eβ) and two posterior distributions (Ez and Ee) as
By setting the quadratic form of \({{\varvec{\upbeta}}} = \left[ {\beta_{1}^{2} ,\beta_{2}^{2} } \right]^{{\text{ T}}}\) to compute the derivative and to ensure its value positive, the derivative in Eq. (32) [or Eq. (16)] can be estimated as
where F2* and F3*denote the F2 and F3 terms without β1 and β2.
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Guo, Q., Ba, J. & Carcione, J.M. Multi-Objective Petrophysical Seismic Inversion Based on the Double-Porosity Biot–Rayleigh Model. Surv Geophys 43, 1117–1141 (2022). https://doi.org/10.1007/s10712-022-09692-6
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DOI: https://doi.org/10.1007/s10712-022-09692-6