Azimuthal Seismic Amplitude Variation with Offset and Azimuth Inversion in Weakly Anisotropic Media with Orthorhombic Symmetry

Abstract

Seismic amplitude variation with offset and azimuth (AVOaz) inversion is well known as a popular and pragmatic tool utilized to estimate fracture parameters. A single set of vertical fractures aligned along a preferred horizontal direction embedded in a horizontally layered medium can be considered as an effective long-wavelength orthorhombic medium. Estimation of Thomsen’s weak-anisotropy (WA) parameters and fracture weaknesses plays an important role in characterizing the orthorhombic anisotropy in a weakly anisotropic medium. Our goal is to demonstrate an orthorhombic anisotropic AVOaz inversion approach to describe the orthorhombic anisotropy utilizing the observable wide-azimuth seismic reflection data in a fractured reservoir with the assumption of orthorhombic symmetry. Combining Thomsen’s WA theory and linear-slip model, we first derive a perturbation in stiffness matrix of a weakly anisotropic medium with orthorhombic symmetry under the assumption of small WA parameters and fracture weaknesses. Using the perturbation matrix and scattering function, we then derive an expression for linearized PP-wave reflection coefficient in terms of P- and S-wave moduli, density, Thomsen’s WA parameters, and fracture weaknesses in such an orthorhombic medium, which avoids the complicated nonlinear relationship between the orthorhombic anisotropy and azimuthal seismic reflection data. Incorporating azimuthal seismic data and Bayesian inversion theory, the maximum a posteriori solutions of Thomsen’s WA parameters and fracture weaknesses in a weakly anisotropic medium with orthorhombic symmetry are reasonably estimated with the constraints of Cauchy a priori probability distribution and smooth initial models of model parameters to enhance the inversion resolution and the nonlinear iteratively reweighted least squares strategy. The synthetic examples containing a moderate noise demonstrate the feasibility of the derived orthorhombic anisotropic AVOaz inversion method, and the real data illustrate the inversion stabilities of orthorhombic anisotropy in a fractured reservoir.

Appendices

Appendix 1: Derivation for Linearized PP-Wave Reflection Coefficient in a Weakly Anisotropic Medium with Orthorhombic Symmetry Formed by a Single Set of Aligned Vertical Fractures Embedded in a VTI Background

The relationships between the subscripts (I, J) and (i, j, k, l) in Eq. (30) are given by Shaw and Sen (2006)

$$I = i\delta_{ij} + \left( {9 - i - j} \right)\left( {1 - \delta_{ij} } \right),$$

(41)

and

$$J = k\delta_{kl} + \left( {9 - k - l} \right)\left( {1 - \delta_{kl} } \right),$$

(42)

where $\delta_{ij}$ and $\delta_{kl}$ both denote the Kronecker delta.

For the case of P-wave incidence and reflection, the polarization and slowness vectors are given by

$$t\text{ = }\left[ {\sin \theta \cos \phi ,\sin \theta \sin \phi ,\cos \theta } \right],$$

(43)

$$t^{\prime } = \left[ { - \sin \theta \cos \phi , - \sin \theta \sin \phi ,\cos \theta } \right],$$

(44)

$$p = {1 \mathord{\left/ {\vphantom {1 \alpha }} \right. \kern-0pt} \alpha }\left[ {\sin \theta \cos \phi ,\sin \theta \sin \phi ,\cos \theta } \right],$$

(45)

and

$$p^{\prime } = {1 \mathord{\left/ {\vphantom {1 \alpha }} \right. \kern-0pt} \alpha }\left[ { - \sin \theta \cos \phi , - \sin \theta \sin \phi ,\cos \theta } \right],$$

(46)

where α represents the background P-wave velocity, and $\phi$ represents the azimuthal phase angle.

Substituting Eqs. (43) and (44) into Eq. (30), the expression of $\xi$ is then given by

$$\xi = \cos^{2} \theta - \sin^{2} \theta = \cos 2\theta .$$

(47)

Substituting Eqs. (43)–(47) into Eq. (30), the expression of $\eta_{IJ}$ is then given by

$$\begin{aligned} \eta_{11} & = {{\sin^{4} \theta \cos^{4} \phi } \mathord{\left/ {\vphantom {{\sin^{4} \theta \cos^{4} \phi } {\alpha^{2} }}} \right. \kern-0pt} {\alpha^{2} }},\quad \eta_{12} = {{\sin^{4} \theta \sin^{2} \phi \cos^{2} \phi } \mathord{\left/ {\vphantom {{\sin^{4} \theta \sin^{2} \phi \cos^{2} \phi } {\alpha^{2} }}} \right. \kern-0pt} {\alpha^{2} }},\quad \eta_{13} = {{\sin^{2} \theta \cos^{2} \theta \cos^{2} \phi } \mathord{\left/ {\vphantom {{\sin^{2} \theta \cos^{2} \theta \cos^{2} \phi } {\alpha^{2} }}} \right. \kern-0pt} {\alpha^{2} }}, \\ \eta_{22} & = {{\sin^{4} \theta \sin^{4} \phi } \mathord{\left/ {\vphantom {{\sin^{4} \theta \sin^{4} \phi } {\alpha^{2} }}} \right. \kern-0pt} {\alpha^{2} }},\quad \eta_{23} = {{\sin^{2} \theta \cos^{2} \theta \sin^{2} \phi } \mathord{\left/ {\vphantom {{\sin^{2} \theta \cos^{2} \theta \sin^{2} \phi } {\alpha^{2} }}} \right. \kern-0pt} {\alpha^{2} }},\quad \eta_{33} = {{\cos^{4} \theta } \mathord{\left/ {\vphantom {{\cos^{4} \theta } {\alpha^{2} }}} \right. \kern-0pt} {\alpha^{2} }}, \\ \eta_{44} & = {{ - 4\sin^{2} \theta \cos^{2} \theta \sin^{2} \phi } \mathord{\left/ {\vphantom {{ - 4\sin^{2} \theta \cos^{2} \theta \sin^{2} \phi } {\alpha^{2} }}} \right. \kern-0pt} {\alpha^{2} }},\quad \eta_{55} = {{ - 4\sin^{2} \theta \cos^{2} \theta \cos^{2} \phi } \mathord{\left/ {\vphantom {{ - 4\sin^{2} \theta \cos^{2} \theta \cos^{2} \phi } {\alpha^{2} }}} \right. \kern-0pt} {\alpha^{2} }},\text{ } \\ \eta_{66} & = {{4\sin^{4} \theta \sin^{2} \phi \cos^{2} \phi } \mathord{\left/ {\vphantom {{4\sin^{4} \theta \sin^{2} \phi \cos^{2} \phi } {\alpha^{2} }}} \right. \kern-0pt} {\alpha^{2} }},\quad \eta_{21} = \eta_{12} \text{,}\quad \eta_{31} = \eta_{13} \text{,}\quad \eta_{32} = \eta_{23} . \\ \end{aligned}$$

(48)

The calculation of Eq. (30) then yields

$$\begin{aligned} S & = \Delta \rho \xi + \sum\limits_{I = 1}^{6} {\sum\limits_{J = 1}^{6} {\Delta C_{IJ} \eta_{IJ} } } \\ & = \Delta \rho \cos 2\theta + \frac{{\sin^{4} \theta \cos^{4} \phi }}{{\alpha^{2} }}\left[ {\Delta M - M\Delta \delta_{\text{N}} + 2M\Delta \varepsilon_{\text{b}} } \right] \\ & \quad +\, \frac{{2\sin^{4} \theta \sin^{2} \phi \cos^{2} \phi }}{{\alpha^{2} }}\left[ {\Delta \lambda - \lambda \Delta \delta_{\text{N}} + 2M\Delta \varepsilon_{\text{b}} - 4\mu \Delta \gamma_{\text{b}} } \right] \\ & \quad + \,\frac{{2\sin^{2} \theta \cos^{2} \theta \cos^{2} \phi }}{{\alpha^{2} }}\left[ {\Delta \lambda - \lambda \Delta \delta_{\text{N}} + M\Delta \delta_{\text{b}} } \right] + \frac{{\sin^{4} \theta \sin^{4} \phi }}{{\alpha^{2} }}\left[ {\Delta M - M\chi^{2} \Delta \delta_{\text{N}} + 2M\Delta \varepsilon_{\text{b}} } \right] \\ & \quad + \,\frac{{2\sin^{2} \theta \cos^{2} \theta \sin^{2} \phi }}{{\alpha^{2} }}\left[ {\Delta \lambda - \lambda \chi \Delta \delta_{\text{N}} + M\Delta \delta_{\text{b}} } \right] \\ & \quad +\, \frac{{\cos^{4} \theta }}{{\alpha^{2} }}\left[ {\Delta M - M\chi^{2} \Delta \delta_{\text{N}} } \right]- \frac{{4\sin^{2} \theta \cos^{2} \theta \sin^{2} \phi }}{{\alpha_{0}^{2} }}\Delta \mu \\ & \quad - \frac{{4\sin^{2} \theta \cos^{2} \theta \cos^{2} \phi }}{{\alpha^{2} }}\left[ {\Delta \mu - \mu \Delta \delta_{\text{V}} } \right] + \frac{{4\sin^{4} \theta \sin^{2} \phi \cos^{2} \phi }}{{\alpha^{2} }}\left[ {\Delta \mu - \mu \Delta \delta_{\text{H}} - 2\mu \Delta \gamma_{\text{b}} } \right] \\ & = \frac{1}{{\alpha^{2} }}\Delta M - \frac{{\sin^{2} \theta \cos^{2} \theta }}{{\alpha^{2} }}\Delta \mu + \cos 2\theta \Delta \rho + \frac{{2M\sin^{4} \theta }}{{\alpha^{2} }}\Delta \varepsilon_{\text{b}} + \frac{{2M\sin^{2} \theta \cos^{2} \theta }}{{\alpha^{2} }}\Delta \delta_{\text{b}} \\ & \quad - \,\frac{M}{{\alpha^{2} }}\left[ {2\frac{\mu }{M}\left( {\sin^{2} \theta \sin^{2} \phi + \cos^{2} \theta } \right) - 1} \right]^{2} \Delta \delta_{\text{N}} + \frac{{4\mu \sin^{2} \theta \cos^{2} \phi }}{{\alpha^{2} }}\Delta \delta_{\text{V}} \\ & \quad-\, \frac{{4\mu \sin^{4} \theta \sin^{2} \phi \cos^{2} \phi }}{{\alpha^{2} }}\Delta \delta_{\text{H}} . \\ \end{aligned}$$

(49)

Combining Eq. (49), the calculation of Eq. (29) finally gives

$$\begin{aligned} R_{\text{PP}} \left( {\theta ,\phi } \right) & = \frac{{\sec^{2} \theta }}{4M}\Delta M - \frac{{2\sin^{2} \theta }}{M}\Delta \mu + \frac{1}{2\rho }\left( {1 - \frac{{\sec^{2} \theta }}{2}} \right)\Delta \rho \\ & \quad + \,\frac{{\sin^{2} \theta \tan^{2} \theta }}{2}\Delta \varepsilon_{\text{b}} + \frac{{\sin^{2} \theta }}{2}\Delta \delta_{\text{b}} - \frac{{\sec^{2} \theta }}{4}\left[ {2\frac{\mu }{M}\left( {\sin^{2} \theta \sin^{2} \phi + \cos^{2} \theta } \right) - 1} \right]^{2} \Delta \delta_{\text{N}} \\ & \quad + \,\frac{\mu }{M}\sin^{2} \theta \cos^{2} \phi \Delta \delta_{\text{V}} - \frac{\mu }{M}\sin^{2} \theta \tan^{2} \theta \sin^{2} \phi \cos^{2} \phi \Delta \delta_{\text{H}} , \\ \end{aligned}$$

(50)

Appendix 2: Derivation for Linearized PP-Wave Reflection Coefficients in a Weakly Anisotropic Medium with Orthorhombic Symmetry Formed by Two Orthogonal Vertical Fracture Sets Embedded in an Isotropic or VTI Background

Two orthogonal vertical fracture sets embedded in an isotropic or VTI background can be both considered as an effective long-wavelength orthorhombic medium (Bakulin et al. 2000, 2002). To further simplify the procedure of parameter estimation, the fracture sets are assumed to be rotationally invariant. For such a weakly anisotropic medium with orthorhombic symmetry formed by two orthogonal vertical fracture sets embedded in an isotropic background, where the first fracture set is perpendicular to the x axis (shown in Fig. 1b), neglecting the terms that contain $\delta_{{{\text{N}}1}} \delta_{{{\text{N}}2}}$ and $\delta_{{{\text{T}}1}} \delta_{{{\text{T}}2}}$ for the case of small fracture weaknesses, an expression for the effective elastic stiffness tensor can be approximated as

$${\mathbf{C}}_{\text{OA}} = \left[ {\begin{array}{*{20}c} {C_{11} } & {C_{12} } & {C_{13} } & 0 & 0 & 0 \\ {C_{12} } & {C_{22} } & {C_{23} } & 0 & 0 & 0 \\ {C_{13} } & {C_{23} } & {C_{33} } & 0 & 0 & 0 \\ 0 & 0 & 0 & {C_{44} } & 0 & 0 \\ 0 & 0 & 0 & 0 & {C_{55} } & 0 \\ 0 & 0 & 0 & 0 & 0 & {C_{66} } \\ \end{array} } \right] \approx \left[ {\begin{array}{*{20}c} {{\hat{\mathbf{C}}}_{1} } & {\mathbf{0}} \\ {\mathbf{0}} & {{\hat{\mathbf{C}}}_{2} } \\ \end{array} } \right],$$

(51)

where 0 represents the 3 × 3 zero matrix, and ${{\hat{\mathbf{C}}}}_{1}$ and ${{\hat{\mathbf{C}}}}_{2}$ are given by

$${{\hat{\mathbf{C}}}}_{1} = \left[ {\begin{array}{*{20}l} {M\left( {1 - \delta_{{{\text{N}}1}} - \chi^{2} \delta_{{{\text{N}}2}} } \right)} \hfill & {\lambda \left( {1 - \delta_{{{\text{N}}1}} - \delta_{{{\text{N}}2}} } \right)} \hfill & {\lambda \left( {1 - \delta_{{{\text{N}}1}} - \chi \delta_{{{\text{N}}2}} } \right)} \hfill \\ {\lambda \left( {1 - \delta_{{{\text{N}}1}} - \delta_{{{\text{N}}2}} } \right)} \hfill & {M\left( {1 - \chi^{2} \delta_{{{\text{N}}1}} - \delta_{{{\text{N}}2}} } \right)} \hfill & {\lambda \left( {1 - \chi \delta_{{{\text{N}}1}} - \delta_{{{\text{N}}2}} } \right)} \hfill \\ {\lambda \left( {1 - \delta_{{{\text{N}}1}} - \chi \delta_{{{\text{N}}2}} } \right)} \hfill & {\lambda \left( {1 - \chi \delta_{{{\text{N}}1}} - \delta_{{{\text{N}}2}} } \right)} \hfill & {M\left( {1 - \chi^{2} \delta_{{{\text{N}}1}} - \chi^{2} \delta_{{{\text{N}}2}} } \right)} \hfill \\ \end{array} } \right],$$

(52)

and

$${{\hat{\mathbf{C}}}}_{2} = \left[ {\begin{array}{*{20}c} {\mu \left( {1 - \delta_{{{\text{T}}2}} } \right)} & 0 & 0 \\ 0 & {\mu \left( {1 - \delta_{{{\text{T}}1}} } \right)} & 0 \\ 0 & 0 & {\mu \left( {1 - \delta_{{{\text{T}}1}} - \delta_{{{\text{T}}2}} } \right)} \\ \end{array} } \right].$$

(53)

Here $\delta_{{{\text{N}}i}} = {{Z_{{{\text{N}}i}} M} \mathord{\left/ {\vphantom {{Z_{{{\text{N}}i}} M} {\left[ {1 + Z_{{{\text{N}}i}} M} \right]}}} \right. \kern-0pt} {\left[ {1 + Z_{{{\text{N}}i}} M} \right]}}$ and $\delta_{{{\text{T}}i}} = {{Z_{{{\text{T}}i}} \mu } \mathord{\left/ {\vphantom {{Z_{{{\text{T}}i}} \mu } {\left[ {1 + Z_{{{\text{T}}i}} \mu } \right]}}} \right. \kern-0pt} {\left[ {1 + Z_{{{\text{T}}i}} \mu } \right]}}$ represent the normal and tangential weaknesses of two orthogonal vertical fracture sets related to the corresponding normal and tangential fracture compliances Z _Ni and Z _Ti .

Following a similar derivation method for the case of small fracture weaknesses, the expression for linearized PP-wave reflection coefficient in such an orthorhombic medium can be expressed as

$$\begin{aligned} R_{\text{PP}} \left( {\theta ,\phi } \right) & = \frac{{\sec^{2} \theta }}{4}\frac{\Delta M}{M} - 2g\sin^{2} \theta \frac{\Delta \mu }{\mu } + \left( {\frac{1}{2} - \frac{{\sec^{2} \theta }}{4}} \right)\frac{\Delta \rho }{\rho } \\ & \quad - \frac{{\sec^{2} \theta }}{4}\left[ {2g\left( {\sin^{2} \theta \sin^{2} \phi + \cos^{2} \theta } \right) - 1} \right]^{2} \Delta \delta_{{{\text{N}}1}} \\ & \quad+ g\sin^{2} \theta \cos^{2} \phi \left( {1 - \tan^{2} \theta \sin^{2} \phi } \right)\Delta \delta_{{{\text{T}}1}} \\ & \quad - \frac{{\sec^{2} \theta }}{4}\left[ {2g\left( {\sin^{2} \theta \cos^{2} \phi + \cos^{2} \theta } \right) - 1} \right]^{2} \Delta \delta_{{{\text{N}}2}} \\ & \quad+ g\sin^{2} \theta \sin^{2} \phi \left( {1 - \tan^{2} \theta \cos^{2} \phi } \right)\Delta \delta_{{{\text{T}}2}} . \\ \end{aligned}$$

(54)

Similarly, for such an orthorhombic medium formed by two orthogonal vertical fracture sets embedded in a VTI background (shown in Fig. 1c), neglecting the terms that contain $\varepsilon_{\text{b}} \delta_{{{\text{N}}1}}$, $\varepsilon_{\text{b}} \delta_{{{\text{N}}2}}$, $\varepsilon_{\text{b}} \delta_{{{\text{N}}1}} \delta_{{{\text{N}}2}}$, $\gamma_{\text{b}} \delta_{{{\text{N}}1}}$, $\gamma_{\text{b}} \delta_{{{\text{N}}2}}$, $\gamma_{\text{b}} \delta_{{{\text{N}}1}} \delta_{{{\text{N}}2}}$,$\delta_{\text{b}} \delta_{{{\text{N}}1}}$, $\delta_{\text{b}} \delta_{{{\text{N}}2}}$, $\delta_{\text{b}} \delta_{{{\text{N}}1}} \delta_{{{\text{N}}2}}$, $\gamma_{\text{b}} \delta_{{{\text{T}}1}}$, $\gamma_{\text{b}} \delta_{{{\text{T}}2}}$, and $\gamma_{\text{b}} \delta_{{{\text{T}}1}} \delta_{{{\text{T}}2}}$ for the case of weak anisotropy and small fracture weaknesses, an expression for the effective elastic stiffness tensor can be approximated as

$${\mathbf{C}}_{\text{OA}} = \left[ {\begin{array}{llllll} {C_{11} } & {C_{12} } & {C_{13} } & 0 & 0 & 0 \\ {C_{12} } & {C_{22} } & {C_{23} } & 0 & 0 & 0 \\ {C_{13} } & {C_{23} } & {C_{33} } & 0 & 0 & 0 \\ 0 & 0 & 0 & {C_{44} } & 0 & 0 \\ 0 & 0 & 0 & 0 & {C_{55} } & 0 \\ 0 & 0 & 0 & 0 & 0 & {C_{66} } \\ \end{array} } \right] \approx \left[ {\begin{array}{*{20}c} {\hat{\hat{{\mathbf{C}}}}_{1} } & {\mathbf{0}} \\ {\mathbf{0}} & {\hat{\hat{{\mathbf{C}}}}_{2} } \\ \end{array} } \right],$$

(55)

where 0 represents the 3 × 3 zero matrix, and ${\hat{\hat{\mathbf{C}}}}_{1}$ and ${\hat{\hat{\mathbf{C}}}}_{2}$ are given by

$${\hat{\hat{\mathbf{C}}}}_{1} = \left[ {\begin{array}{*{20}l} {M\left( {1 - \delta_{{{\text{N}}1}} - \chi^{2} \delta_{{{\text{N}}2}} } \right) + 2M\varepsilon_{\text{b}} } \hfill & {\lambda \left( {1 - \delta_{{{\text{N}}1}} - \delta_{{{\text{N}}2}} } \right) + 2M\varepsilon_{\text{b}} - 4\mu \gamma_{\text{b}} } \hfill & {\lambda \left( {1 - \delta_{{{\text{N}}1}} - \chi \delta_{{{\text{N}}2}} } \right) + M\delta_{\text{b}} } \hfill \\ {\lambda \left( {1 - \delta_{{{\text{N}}1}} - \delta_{{{\text{N}}2}} } \right) + 2M\varepsilon_{\text{b}} - 4\mu \gamma_{\text{b}} } \hfill & {M\left( {1 - \chi^{2} \delta_{{{\text{N}}1}} - \delta_{{{\text{N}}2}} } \right) + 2M\varepsilon_{\text{b}} } \hfill & {\lambda \left( {1 - \chi \delta_{{{\text{N}}1}} - \delta_{{{\text{N}}2}} } \right) + M\delta_{\text{b}} } \hfill \\ {\lambda \left( {1 - \delta_{{{\text{N}}1}} - \chi \delta_{{{\text{N}}2}} } \right) + M\delta_{\text{b}} } \hfill & {\lambda \left( {1 - \chi \delta_{{{\text{N}}1}} - \delta_{{{\text{N}}2}} } \right) + M\delta_{\text{b}} } \hfill & {M\left( {1 - \chi^{2} \delta_{{{\text{N}}1}} - \chi^{2} \delta_{{{\text{N}}2}} } \right)} \hfill \\ \end{array} } \right],$$

(56)

and

$${\hat{\hat{\mathbf{C}}}}_{2} = \left[ {\begin{array}{*{20}c} {\mu \left( {1 - \Delta \delta_{{{\text{T}}2}} } \right)} & 0 & 0 \\ 0 & {\mu \left( {1 - \delta_{{{\text{T}}1}} } \right)} & 0 \\ 0 & 0 & {\mu \left( {1 - \delta_{{{\text{T}}1}} - \delta_{{{\text{T}}2}} } \right) - 2\mu \gamma_{\text{b}} } \\ \end{array} } \right].$$

(57)

Following a similar derivation method for the case of weak anisotropy and small fracture weaknesses, the expression for linearized PP-wave reflection coefficient in such an orthorhombic medium can be expressed as

$$\begin{aligned} R_{\text{PP}} \left( {\theta ,\phi } \right) & = \frac{{\sec^{2} \theta }}{4}\frac{\Delta M}{M} - 2g\sin^{2} \theta \frac{\Delta \mu }{\mu } + \left( {\frac{1}{2} - \frac{{\sec^{2} \theta }}{4}} \right)\frac{\Delta \rho }{\rho } + \frac{{\sin^{2} \theta \tan^{2} \theta }}{2}\Delta \varepsilon_{\text{b}} + \frac{{\sin^{2} \theta }}{2}\Delta \delta_{\text{b}} \\ & \quad - \,\frac{{\sec^{2} \theta }}{4}\left[ {2g\left( {\sin^{2} \theta \sin^{2} \phi + \cos^{2} \theta } \right) - 1} \right]^{2} \Delta \delta_{{{\text{N}}1}} \\ & \quad+ g\sin^{2} \theta \cos^{2} \phi \left( {1 - \tan^{2} \theta \sin^{2} \phi } \right)\Delta \delta_{{{\text{T}}1}} \\ & \quad - \,\frac{{\sec^{2} \theta }}{4}\left[ {2g\left( {\sin^{2} \theta \cos^{2} \phi + \cos^{2} \theta } \right) - 1} \right]^{2} \Delta \delta_{{{\text{N}}2}} \\ & \quad+ g\sin^{2} \theta \sin^{2} \phi \left( {1 - \tan^{2} \theta \cos^{2} \phi } \right)\Delta \delta_{{{\text{T}}2}} . \\ \end{aligned}$$

(58)