Model Parameterization and PP-Wave Amplitude Versus Angle and Azimuth (AVAZ) Direct Inversion for Fracture Quasi-Weaknesses in Weakly Anisotropic Elastic Media

Abstract

Homogeneous isotropic or vertically transverse isotropic rocks containing a single set of aligned, vertical fractures exhibits an effective long-wavelength horizontally transverse isotropy (HTI) or orthorhombic anisotropy. The estimation for properties of subsurface fractures has significant application in characterization of naturally fractured rocks. The purpose of this work is to demonstrate an approach of amplitude versus angle and azimuth (AVAZ) direct inversion for fracture characterization utilizing the observable wide-azimuth seismic reflection data in weakly anisotropic elastic media. The simplest single fracture system is HTI model. Much attention has been devoted to the weak-contrast and weak-anisotropy HTI model due to its significance for reservoir characterization. Treating the fractures as linear-slip interfaces, we begin with the derivation for perturbations of stiffness matrix at a planar weak-contrast interface separating two weakly anisotropic HTI half-spaces that share the same fracture normal, as a function of background elastic moduli and fracture parameters. Using the perturbation matrix and scattering function, we then derive a linearized PP-wave reflection coefficient of a weakly HTI medium in terms of P- and S-wave moduli, density, and fracture weaknesses, which builds a linearized relationship between the fracture parameters and reflection coefficient with the priority calculation for the azimuth of fracture normal based on the least square ellipse fitting method. Finally, we reformulate the reflectivity caused by weakness differences to parameterize the weaknesses for the so-called quasi-weaknesses and propose a method of Bayesian AVAZ direct inversion in seismic detection of subsurface fractures. Cauchy and Gaussian probability distribution are used for the a priori information of model parameters and the likelihood function, and the maximum a posteriori estimate of quasi-weaknesses is reasonably estimated with the nonlinear iteratively reweighted least squares algorithm. Synthetic and real data illustrate the applicability of the proposed AVAZ inversion method in fracture characterization.

Appendices

Appendix A: Linearized PP-Wave Reflection Coefficient in a Weakly HTI Medium

For the case of P-wave incidence and reflection, the polarization and slowness vectors are given by (Shaw and Sen 2006)

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

(A-1)

$$t^{'} = \left[ { - \sin \theta \cos \phi , - \sin \theta \sin \phi ,\cos \theta } \right],$$

(A-2)

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

(A-3)

and

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

(A-4)

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

The expression for $\xi$ and $\eta_{mn}$ is then given by (Shaw and Sen 2006)

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

(A-5)

and

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

(A-6)

Combining Eq. (A-6) and Eqs. (8)–(12), the calculation of Eq. (13) yields

$$\begin{aligned} & R_{\text{PP}} = \frac{\Delta \rho }{{4\rho_{b} }}\left( {\xi \sec^{2} \theta } \right) + \sum\limits_{m = 1}^{6} {\sum\limits_{n = 1}^{6} {\frac{{\Delta c_{mn} }}{{4\rho_{b} }}\left( {\eta_{mn} \sec^{2} \theta } \right)} } \\ & = \frac{{\sec^{2} \theta }}{{4\rho_{b} }}\left\{ \begin{aligned} \Delta \rho \cos 2\theta + \frac{{\sin^{4} \theta \cos^{4} \phi }}{{\alpha_{b}^{2} }}\left[ {\Delta M - M_{b} \Delta \delta_{N} } \right]\text{ } + \frac{{2\sin^{4} \theta \sin^{2} \phi \cos^{2} \phi }}{{\alpha_{b}^{2} }}\left[ {\Delta \lambda - \lambda_{b} \Delta \delta_{N} } \right] \hfill \\ \text{ }{ + }\frac{{2\sin^{2} \theta \cos^{2} \theta \cos^{2} \phi }}{{\alpha_{b}^{2} }}\left[ {\Delta \lambda - \lambda_{b} \Delta \delta_{N} } \right] + \frac{{\sin^{4} \theta \sin^{4} \phi }}{{\alpha_{b}^{2} }}\left[ {\Delta M - M_{b} \chi^{2} \Delta \delta_{N} } \right] \hfill \\ \text{ } + \frac{{2\sin^{2} \theta \cos^{2} \theta \sin^{2} \phi }}{{\alpha_{b}^{2} }}\left[ {\Delta \lambda - \lambda_{b} \chi \Delta \delta_{N} } \right] + \frac{{\cos^{4} \theta }}{{\alpha_{b}^{2} }}\left[ {\Delta M - M_{b} \chi^{2} \Delta \delta_{N} } \right] \hfill \\ \text{ } - \frac{{4\sin^{2} \theta \cos^{2} \theta \sin^{2} \phi }}{{\alpha_{b}^{2} }}\Delta \mu \hfill \\ \text{ } - \frac{{4\sin^{2} \theta \cos^{2} \theta \cos^{2} \phi }}{{\alpha_{b}^{2} }}\left[ {\Delta \mu - \mu_{b} \Delta \delta_{T} } \right] + \frac{{4\sin^{4} \theta \sin^{2} \phi \cos^{2} \phi }}{{\alpha_{b}^{2} }}\left[ {\Delta \mu - \mu_{b} \Delta \delta_{T} } \right] \hfill \\ \end{aligned} \right\}, \\ \end{aligned}$$

(A-7)

Therefore, the linearized PP-wave reflection coefficient in a weakly HTI medium can be further deduced as

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

(A-7)

Appendix B: Linearized PP-Wave Reflection Coefficient in a Weakly Orthorhombic Medium

A system of aligned vertical fractures embedded in a VTI background can be considered as an effective long-wavelength orthorhombic medium. In the case that fracture faces are perpendicular to the x1-axis, the effective elastic stiffness tensor ${\mathbf{c}}_{\text{OA}}$ in such an orthorhombic medium can be expressed as (Schoenberg and Helbig 1997)

$${\mathbf{c}}_{\text{OA}} = \left[ {\begin{array}{*{20}c} {c_{11b} \left( {1 - \delta_{N} } \right)} & {c_{12b} \left( {1 - \delta_{N} } \right)} & {c_{13b} \left( {1 - \delta_{N} } \right)} & 0 & 0 & 0 \\ {c_{12b} \left( {1 - \delta_{N} } \right)} & {c_{11b} \left( {1 - \delta_{N} \frac{{C_{12b}^{2} }}{{C_{11b}^{2} }}} \right)} & {c_{13b} \left( {1 - \delta_{N} \frac{{C_{12b} }}{{C_{11b} }}} \right)} & 0 & 0 & 0 \\ {c_{13b} \left( {1 - \delta_{N} } \right)} & {c_{13b} \left( {1 - \delta_{N} \frac{{C_{12b} }}{{C_{11b} }}} \right)} & {c_{33b} \left( {1 - \delta_{N} \frac{{C_{13b}^{2} }}{{C_{11b} C_{33b} }}} \right)} & 0 & 0 & 0 \\ 0 & 0 & 0 & {c_{44b} } & 0 & 0 \\ 0 & 0 & 0 & 0 & {c_{44b} \left( {1 - \delta_{V} } \right)} & 0 \\ 0 & 0 & 0 & 0 & 0 & {c_{66b} \left( {1 - \delta_{H} } \right)} \\ \end{array} } \right],$$

(B-1)

where $c_{ijb}$ represents the stiffness components of a VTI background medium, $\delta_{\text{N}}$, $\delta_{\text{V}}$, and $\delta_{\text{H}}$ represent the dimensionless normal, vertical, and horizontal tangential fracture weaknesses, respectively.

Under the weak-anisotropy assumption, the dimensionless Thomsen’s (1986) weak-anisotropy parameters can be written in terms of stiffness components of VTI background as (Pšenčik and Gajewski 1998; Pšencik and Vavrycuk 1998)

$$\varepsilon_{b} = \frac{{c_{11b} - c_{33b} }}{{2c_{33b} }},$$

(B-2)

$$\text{ }\gamma_{b} = \frac{{c_{66b} - c_{44b} }}{{2c_{44b} }},$$

(B-3)

and

$$\delta_{b} = \frac{{c_{13b} + 2c_{44b} - c_{33b} }}{{c_{33b} }},$$

(B-4)

where $\varepsilon_{b}$, $\text{ }\gamma_{b}$, and $\delta_{b}$ represent the three weak-anisotropy parameters of VTI background.

Under the assumption of small weak-anisotropy and weakness parameters, we neglected the terms that contain $\delta_{N}^{2}$, $\varepsilon_{b}^{2}$, $\gamma_{b}^{2}$, $\delta_{b}^{2}$, $\varepsilon_{b} \delta_{N}$, $\gamma_{b} \delta_{N}$, $\delta_{b} \delta_{N}$, and $\gamma_{b} \delta_{H}$, and derive a new expression for stiffness tensor in terms of weak anisotropic parameters of VTI background and fracture weaknesses in a weakly orthorhombic medium, which is given by

$${\mathbf{c}}_{\text{OA}} = \left[ {\begin{array}{*{20}c} {M_{b} \left( {1 - \delta_{N} + 2\varepsilon_{b} } \right)} & {\left( \begin{aligned} \lambda_{b} \left( {1 - \delta_{N} } \right) + \hfill \\ 2M_{b} \left( {\varepsilon_{b} - 2g\gamma_{b} } \right) \hfill \\ \end{aligned} \right)} & {\lambda_{b} \left( {1 - \delta_{N} } \right) + M_{b} \delta_{b} } & 0 & 0 & 0 \\ {\left( \begin{aligned} \lambda_{b} \left( {1 - \delta_{N} } \right) + \hfill \\ 2M_{b} \left( {\varepsilon_{b} - 2g\gamma_{b} } \right) \hfill \\ \end{aligned} \right)} & {M_{b} \left( {1 - \chi^{2} \delta_{N} } \right) + 2M_{b} \varepsilon_{b} } & {\lambda_{b} \left( {1 - \chi \delta_{N} } \right) + M_{b} \delta_{b} } & 0 & 0 & 0 \\ {\lambda_{b} \left( {1 - \delta_{N} } \right) + M_{b} \delta_{b} } & {\lambda_{b} \left( {1 - \chi \delta_{N} } \right) + M_{b} \delta_{b} } & {M_{b} \left( {1 - \chi^{2} \delta_{N} } \right)} & 0 & 0 & 0 \\ 0 & 0 & 0 & {\mu_{b} } & 0 & 0 \\ 0 & 0 & 0 & 0 & {\mu_{b} \left( {1 - \delta_{V} } \right)} & 0 \\ 0 & 0 & 0 & 0 & 0 & {\mu_{b} \left( {1 - \delta_{H} - 2\gamma_{b} } \right)} \\ \end{array} } \right].$$

(B-5)

Considering the small perturbations in elastic moduli across the interface, and neglecting the terms that contain $\Delta M\delta_{N}$, $\Delta \lambda \delta_{N}$, $\Delta M\varepsilon_{b}$, $\Delta M\delta_{b}$, $\Delta \mu \gamma_{b}$, $\Delta \mu \delta_{V}$, and $\Delta \mu \delta_{H}$ for the case of small weak-anisotropy and weakness parameters, we can derive the perturbations for stiffness components, which are given by

$$\Delta c_{11} \approx \Delta M + 2M_{b} \Delta \varepsilon_{b} - M_{b} \Delta \delta_{N} ,$$

(B-6)

$$\Delta c_{12} \approx \Delta \lambda + 2M_{b} \Delta \varepsilon_{b} - 4\mu_{b} \Delta \gamma_{b} - \lambda_{b} \Delta \delta_{N} ,$$

(B-7)

$$\Delta c_{13} \approx \Delta \lambda + M_{b} \Delta \delta_{b} - \lambda_{b} \Delta \delta_{N} ,$$

(B-8)

$$\Delta c_{22} \approx \Delta M + 2M_{b} \Delta \varepsilon_{b} - M_{b} \chi^{2} \Delta \delta_{N} ,$$

(B-9)

$$\Delta c_{23} \approx \Delta \lambda + M_{b} \Delta \delta_{b} - \lambda_{b} \chi \Delta \delta_{N} ,$$

(B-10)

$$\Delta c_{33} \approx \Delta M - M_{b} \chi^{2} \Delta \delta_{N} ,$$

(B-11)

$$\Delta c_{44} = \Delta \mu ,$$

(B-12)

$$\Delta c_{55} \approx \Delta \mu - \mu_{b} \Delta \delta_{V} ,$$

(B-13)

and

$$\Delta c_{66} \approx \Delta \mu - 2\mu_{b} \Delta \gamma_{b} - \mu_{b} \Delta \delta_{H} ,$$

(B-14)

where $\Delta M$, $\Delta \mu$, $\Delta \lambda$, $\Delta \varepsilon_{b}$, $\Delta \gamma_{b}$, $\Delta \delta_{b}$, $\Delta \delta_{N}$, $\Delta \delta_{V}$, and $\Delta \delta_{H}$ represent the perturbations for elastic moduli, weak-anisotropy parameters of VTI background, and fracture weaknesses between the layers separated by the interface, respectively.

Combining the perturbation matrix and scattering function, we also derive a linearized PP-wave reflection coefficient of a weakly orthorhombic medium, which is given by

$$\begin{aligned} R_{\text{PP}} \left( {\theta ,\phi } \right) & = a_{M} \left( \theta \right)\frac{\Delta M}{{M_{b} }} + a_{\mu } \left( \theta \right)\frac{\Delta \mu }{{\mu_{b} }} + a_{\rho } \left( \theta \right)\frac{\Delta \rho }{{\rho_{b} }} + a_{{\varepsilon_{b} }} \left( \theta \right)\Delta \varepsilon_{b} + a_{{\delta_{b} }} \left( \theta \right)\Delta \delta_{b} \\ & \quad + \;a_{{\delta_{N} }} \left( {\theta ,\phi } \right)\Delta \delta_{N} + a_{{\delta_{V} }} \left( {\theta ,\phi } \right)\Delta \delta_{V} + a_{{\delta_{H} }} \left( {\theta ,\phi } \right)\Delta \delta_{H} , \\ \end{aligned}$$

(B-15)

where $a_{M} \left( \theta \right) = {{\sec^{2} \theta } \mathord{\left/ {\vphantom {{\sec^{2} \theta } 4}} \right. \kern-0pt} 4}$, $a_{\mu } \left( \theta \right) = - 2g\sin^{2} \theta$, $a_{\rho } \left( \theta \right) = {{\left( {1 - {{\sec^{2} \theta } \mathord{\left/ {\vphantom {{\sec^{2} \theta } 2}} \right. \kern-0pt} 2}} \right)} \mathord{\left/ {\vphantom {{\left( {1 - {{\sec^{2} \theta } \mathord{\left/ {\vphantom {{\sec^{2} \theta } 2}} \right. \kern-0pt} 2}} \right)} 2}} \right. \kern-0pt} 2}$, $a_{{\varepsilon_{b} }} \left( \theta \right) = {{\sin^{2} \theta \tan^{2} \theta } \mathord{\left/ {\vphantom {{\sin^{2} \theta \tan^{2} \theta } 2}} \right. \kern-0pt} 2}$, $a_{{\delta_{b} }} \left( \theta \right) = {{\sin^{2} \theta } \mathord{\left/ {\vphantom {{\sin^{2} \theta } 2}} \right. \kern-0pt} 2}$, $a_{{\delta_{N} }} \left( {\theta ,\phi } \right) = - {{\sec^{2} \theta } \mathord{\left/ {\vphantom {{\sec^{2} \theta } 4}} \right. \kern-0pt} 4}\left[ {2g\left( {\sin^{2} \theta \sin^{2} \phi + \cos^{2} \theta } \right) - 1} \right]^{2}$, $a_{{\delta_{V} }} \left( {\theta ,\phi } \right) = g\sin^{2} \theta \cos^{2} \phi$, and $a_{{\delta_{H} }} \left( {\theta ,\phi } \right) = - g\sin^{2} \theta \tan^{2} \theta \sin^{2} \phi \cos^{2} \phi$.