Multi-Objective Petrophysical Seismic Inversion Based on the Double-Porosity Biot–Rayleigh Model

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

• The petrophysical seismic inversion is based on the double-porosity Biot–Rayleigh model

• Spatially varying inclusion volumes are used to describe complex pore structures

• A multi-objective optimization with joint data misfit enables stable results

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

$$ \begin{aligned} & N\nabla^{2} {\mathbf{u}} + \left( {A + N} \right)\nabla \varepsilon + Q_{1} \left( {\zeta^{\left( 1 \right)} + \phi_{c} \varsigma } \right) + Q_{2} \left( {\zeta^{\left( 2 \right)} - \phi_{1} \varsigma } \right) \\ \quad = \rho_{11} {\mathbf{\ddot{u}}} + \rho_{12} {\mathbf{\ddot{U}}}^{\left( 1 \right)} + \rho_{13} {\mathbf{\ddot{U}}}^{\left( 2 \right)} + b_{1} \left( {{\dot{\mathbf{u}}} - {\dot{\mathbf{U}}}^{\left( 1 \right)} } \right) + b_{2} \left( {{\dot{\mathbf{u}}} - {\dot{\mathbf{U}}}^{\left( 2 \right)} } \right), \\ \end{aligned} $$

(19)

$$ Q_{1} \nabla \varepsilon + R_{1} \nabla \left( {\zeta^{\left( 1 \right)} + \phi_{c} \varsigma } \right) = \rho_{12} {\mathbf{\ddot{u}}} + \rho_{22} {\mathbf{\ddot{U}}}^{\left( 1 \right)} - b_{1} \left( {{\dot{\mathbf{u}}} - {\dot{\mathbf{U}}}^{\left( 1 \right)} } \right), $$

(20)

$$ Q_{2} \nabla \varepsilon + R_{2} \nabla \left( {\zeta^{\left( 2 \right)} - \phi_{1} \varsigma } \right) = \rho_{13} {\mathbf{\ddot{u}}} + \rho_{33} {\mathbf{\ddot{U}}}^{\left( 1 \right)} - b_{2} \left( {{\dot{\mathbf{u}}} - {\dot{\mathbf{U}}}^{\left( 2 \right)} } \right), $$

(21)

$$ \begin{gathered} \phi_{c} \left( {Q_{1} \varepsilon + R_{1} \left( {\zeta^{\left( 1 \right)} + \phi_{c} \varsigma } \right)} \right) - \phi_{1} \left( {Q_{2} \varepsilon + R_{2} \left( {\zeta^{\left( 2 \right)} - \phi_{1} \varsigma } \right)} \right) \hfill \\ \, = \frac{1}{3}\rho_{f} \ddot{\varsigma }R_{0}^{2} \frac{{\phi_{1}^{2} \phi_{2} \phi_{20} }}{{\phi_{10} }} + \frac{1}{3}\frac{{\eta \phi_{1}^{2} \phi_{2} \phi_{20} }}{{\kappa_{1} }}\dot{\varsigma }R_{0}^{2} , \hfill \\ \end{gathered} $$

(22)

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 R_o 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

$$ \left| {\begin{array}{*{20}c} {a_{11} k^{2} + b_{11} } & {a_{12} k^{2} + b_{12} } & {a_{13} k^{2} + b_{13} } \\ {a_{21} k^{2} + b_{21} } & {a_{22} k^{2} + b_{22} } & {a_{23} k^{2} + b_{23} } \\ {a_{31} k^{2} + b_{31} } & {a_{32} k^{2} + b_{32} } & {a_{33} k^{2} + b_{33} } \\ \end{array} } \right| = 0, $$

(23)

where

$$ \begin{gathered} a_{11} = A + 2N + {\text{i}}(Q_{2} \phi_{1} - Q_{1} \phi_{2} )x_{1} , \hfill \\ b_{11} = - \rho_{11} \omega^{2} + {\text{i}}\omega \left( {b_{1} + b_{2} } \right), \hfill \\ a_{12} = Q_{1} + {\text{i}}(Q_{2} \phi_{1} - Q_{1} \phi_{2} )x_{2} , \, b_{12} = - \rho_{12} \omega^{2} - {\text{i}}\omega b_{1} , \hfill \\ a_{13} = Q_{2} + {\text{i}}(Q_{2} \phi_{1} - Q_{1} \phi_{2} )x_{3} , \, b_{13} = - \rho_{13} \omega^{2} - {\text{i}}\omega b_{2} , \hfill \\ a_{21} = Q_{1} - {\text{i}}R_{{1}} \phi_{2} x_{3} , \, b_{21} = - \rho_{12} \omega^{2} - {\text{i}}\omega b_{1} , \hfill \\ a_{22} = R_{1} - {\text{i}}R_{{1}} \phi_{2} x_{2} , \, b_{22} = - \rho_{22} \omega^{2} + {\text{i}}\omega b_{1} , \hfill \\ a_{23} = - {\text{i}}R_{{1}} \phi_{2} x_{3} , \, b_{23} = 0, \hfill \\ a_{31} = Q_{2} + {\text{i}}R_{{2}} \phi_{1} x_{1} , \, b_{31} = - \rho_{13} \omega^{2} - {\text{i}}\omega b_{2} , \hfill \\ a_{32} = {\text{i}}R_{{2}} \phi_{1} x_{2} , \, b_{32} = 0, \hfill \\ a_{33} = R_{2} + {\text{i}}R_{{2}} \phi_{1} x_{3} , \, b_{33} = - \rho_{33} \omega^{2} + {\text{i}}\omega b_{2} . \hfill \\ \end{gathered} $$

(24)

and

$$ \begin{gathered} x_{1} = \frac{{{\text{i}}\left( {\phi_{{2}} Q_{1} - \phi_{{1}} Q_{2} } \right)}}{Z}, \, x_{2} = \frac{{{\text{i}}\phi_{{2}} R_{1} }}{Z}, \, x_{3} = - \frac{{{\text{i}}\phi_{{1}} R_{2} }}{Z}, \hfill \\ Z = \frac{{{\text{i}}\omega \eta \phi_{1}^{2} \phi_{2} \phi_{20} R_{0}^{2} }}{{3\kappa_{1} }} - \frac{{\rho_{f} \omega^{2} R_{0}^{2} \phi_{1}^{2} \phi_{2} \phi_{20} }}{{3\phi_{10} }} - \left( {\phi_{2}^{2} R_{1} + \phi_{1}^{2} R_{2} } \right) \, . \hfill \\ \end{gathered} $$

(25)

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

$$ V_{{\text{P}}} = \left[ {{\text{Re}}\left( {\frac{k}{\omega }} \right)} \right]^{ - 1} , $$

(26)

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

$$ P\left( {{\mathbf{d}}_{{{\text{seis}}}} ,{\mathbf{d}}_{{{\text{elas}}}} |{{\varvec{\upbeta}}}} \right) = \sum\limits_{{{\mathbf{z}} \in \Omega }} {P\left( {{\mathbf{d}}_{{{\text{seis}}}} ,{\mathbf{d}}_{{{\text{elas}}}} |{\mathbf{z}},{{\varvec{\upbeta}}}} \right) \times P\left( {{\mathbf{z}}|{{\varvec{\upbeta}}}} \right)} , $$

(27)

where the Ω denotes the data space of z. Given the known β, we have

$$ \begin{gathered} P\left( {{\mathbf{d}}_{{{\text{seis}}}} ,{\mathbf{d}}_{{{\text{elas}}}} |{\mathbf{z}},{{\varvec{\upbeta}}}} \right) = P\left( {{\mathbf{d}}_{{{\text{seis}}}} ,{\mathbf{d}}_{{{\text{elas}}}} |{\mathbf{z}}} \right) \approx P\left( {{\mathbf{d}}_{{{\text{seis}}}} |{\mathbf{z}}} \right) \times P\left( {{\mathbf{d}}_{{{\text{elas}}}} |{\mathbf{z}},{{\varvec{\upbeta}}}} \right) \hfill \\ \, = \frac{1}{{K_{{\text{s}}} }}\exp \left( {\frac{{}}{{}}F_{1} \left( {{\mathbf{d}}_{{{\text{seis}}}} ,{\mathbf{z}}} \right)} \right) \times \frac{1}{{K_{{\text{e}}} }}\exp \left( {\frac{{}}{{}}F_{2} \left( {{\mathbf{d}}_{{{\text{elas}}}} ,{\mathbf{z}},\beta_{1} } \right)} \right), \hfill \\ \end{gathered} $$

(28)

where F₁ and F₂ denote the seismic and elastic misfit terms in Eq. (12), and

$$ K_{{\text{s}}} = \sum\limits_{{{\mathbf{z}} \in \Omega }} {P({\mathbf{d}}_{{{\text{seis}}}} |{\mathbf{z}})} {\text{ and }}K_{{\text{e}}} = \sum\limits_{{{\mathbf{z}} \in \Omega }} {P({\mathbf{d}}_{{{\text{elas}}}} |{\mathbf{z}},\beta_{1} )} $$

(29)

are the normalization constants.

The prior distribution of P(z|β) is

$$ P({\mathbf{z}}|{{\varvec{\upbeta}}}) = \frac{1}{{K_{{\upbeta }} }}\exp \left( {F_{3} \left( {{\mathbf{z}},\beta_{2} } \right)} \right), $$

(30)

where F₃ denotes the prior term in Eq. (12), and

$$ K_{{\upbeta }} = \sum\limits_{{{\mathbf{z}} \in \Omega }} {\exp } \left( {F_{3} ({\mathbf{z}},\beta_{2} )} \right) $$

(31)

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(d_seis,d_elas|β)

$$ L\left( {{\varvec{\upbeta}}} \right) = - \ln \left( {\frac{{K_{{\text{z}}} }}{{K_{{\text{s}}} K_{{\text{e}}} K_{{\upbeta }} }}} \right) = \ln K_{{\text{z}}} - \ln K_{{\text{s}}} - \ln K_{{\text{e}}} - \ln K_{{\upbeta }} , $$

(32)

with

$$ K_{{\text{z}}} = \sum\limits_{{{\mathbf{z}} \in \Omega }} {\exp } \left( {F_{1} \left( {{\mathbf{d}}_{{{\text{seis}}}} ,{\mathbf{z}}} \right)} \right.\;F_{2} \left( {{\mathbf{d}}_{{{\text{elas}}}} ,{\mathbf{z}},\beta_{1} } \right)F_{3} \left( {{\mathbf{z}},\beta_{2} } \right)). $$

(33)

By employing the Gauss–Newton descent method to minimize the log-likelihood function 32, β can be iteratively updated as

$$ {{\varvec{\upbeta}}}_{{}}^{{\left( {n + {1}} \right)}} = {{\varvec{\upbeta}}}_{{}}^{\left( n \right)} - \left[ {\left( {\frac{\partial L}{{\partial {{\varvec{\upbeta}}}}}} \right)^{{\text{T}}} \left( {\frac{\partial L}{{\partial {{\varvec{\upbeta}}}}}} \right)} \right]\frac{\partial L}{{\partial {{\varvec{\upbeta}}}}}, $$

(34)

with

$$ \begin{gathered} \frac{\partial L}{{\partial {{\varvec{\upbeta}}}}} \, = \sum\limits_{{{\mathbf{z}} \in \Omega }} {\left( { - \frac{{\partial F_{2} }}{{\partial {{\varvec{\upbeta}}}}} - \frac{{\partial F_{3} }}{{\partial {{\varvec{\upbeta}}}}}} \right)\frac{1}{{K_{{\text{z}}} }}\exp \left( {F_{1} } \right.\left. {\;F_{2} \;F_{3} } \right)} \hfill \\ \, - \sum\limits_{{{\mathbf{z}} \in \Omega }} {\left( { - \frac{{\partial F_{2} }}{{\partial {{\varvec{\upbeta}}}}}} \right)\frac{1}{{K_{{\text{e}}} }}\exp \left( {F_{2} } \right)} - \sum\limits_{{{\mathbf{z}} \in \Omega }} {\left( { - \frac{{\partial F_{3} }}{{\partial {{\varvec{\upbeta}}}}}} \right)\frac{1}{{K_{{\upbeta }} }}\exp \left( {F_{3} } \right)} . \hfill \\ \end{gathered} $$

(35)

Introducing the expectation of z, regarding its probability distribution, Eq. (35) can be estimated from the expectations of one prior distribution (E_β) and two posterior distributions (E_z and E_e) as

$$ \frac{\partial L}{{\partial {{\varvec{\upbeta}}}}} = - E_{{\text{z}}} \left( {\frac{{\partial F_{2} }}{{\partial {{\varvec{\upbeta}}}}}} \right) - E_{{\text{z}}} \left( {\frac{{\partial F_{3} }}{{\partial {{\varvec{\upbeta}}}}}} \right) + E_{{\text{e}}} \left( {\frac{{\partial F_{2} }}{{\partial {{\varvec{\upbeta}}}}}} \right) + E_{{\upbeta }} \left( {\frac{{\partial F_{3} }}{{\partial {{\varvec{\upbeta}}}}}} \right). $$

(36)

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

$$ \frac{\partial L}{{\partial {{\varvec{\upbeta}}}}} = \left( \begin{gathered} \frac{\partial L}{{\partial \beta_{1} }} \hfill \\ \frac{\partial L}{{\partial \beta_{2} }} \hfill \\ \end{gathered} \right) = \left( \begin{gathered} - 2\beta_{1} \left( {E_{{\text{z}}} \left( {F_{2}^{ * } } \right) - E_{{\text{e}}} \left( {F_{2}^{ * } } \right)} \right) \hfill \\ - 2\beta_{2} \left( {E_{{\text{z}}} \left( {F_{3}^{ * } } \right) - E_{\beta } \left( {F_{3}^{ * } } \right)} \right) \hfill \\ \end{gathered} \right), $$

(37)

where F₂^* and F₃^*denote the F₂ and F₃ terms without β₁ and β₂.