Azimuthal Seismic Amplitude Difference Inversion for Fracture Weakness

Abstract

Fracture weakness prediction is an important task in fractured reservoir analysis. We propose a new method to use seismic amplitude differences between azimuths to estimate the normal and tangential fracture weaknesses under the assumption that the anisotropic perturbation of the reflection coefficient is mainly induced by fractures. We first derive an expression of the reflection coefficient in terms of the normal and tangential fracture weaknesses for the case of an interface separating two fractured media. Then we use the linear fitting method to get the relationship between the two fracture weaknesses, and change the variables to precondition the inversion problem. The Bayesian framework, under the hypothesis of a Cauchy distribution prior information and a Gaussian distribution likelihood function, is employed to construct the objective function, and an initial low-frequency constraint is introduced to the objective function to make the inversion more stable. The conjugate gradient algorithm is adopted to solve the inverse problem. Tests on both synthetic and real data demonstrate that the normal and tangential fracture weaknesses can be estimated reasonably in the case of seismic data containing a moderate noise, and our inversion approach appears to be a stable method for predicting the fracture weaknesses.

Appendix A: The Derivation of PP-Wave Reflection Coefficient

The relationships among m, n, i, j, k and l in Eq. (5) are given by Shaw and Sen (2006)

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

(26)

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

(27)

where $\delta_{{\text{ij}}}$ and $\delta_{{\text{kl}}}$ are Kronecker delta.

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

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

(28)

$$p = \frac{1}{\alpha }\left[ {\sin \theta \cos \phi ,\sin \theta \sin \phi ,\cos \theta } \right],\quad p^{\prime } = \frac{1}{\alpha }\left[ { - \sin \theta \cos \phi , - \sin \theta \sin \phi ,\cos \theta } \right],$$

(29)

The expression of η is given by

$$\begin{aligned} \eta_{11} = \frac{{\sin^{4} \theta \cos^{4} \phi }}{{\alpha^{2} }},\;\eta_{12} = \frac{{\sin^{4} \theta \sin^{2} \phi \cos^{2} \phi }}{{\alpha^{2} }},\;\eta_{13} = \frac{{\sin^{2} \theta \cos^{2} \theta \cos^{2} \phi }}{{\alpha^{2} }},\;\eta_{ 2 1} = \eta_{ 1 2} , \hfill \\ \eta_{22} = \frac{{\sin^{4} \theta \sin^{4} \phi }}{{\alpha^{2} }},\;\eta_{23} = \frac{{\sin^{2} \theta \cos^{2} \theta \sin^{2} \phi }}{{\alpha^{2} }},\;\eta_{ 3 1} = \eta_{ 1 3} ,\eta_{ 3 2} = \eta_{ 2 3} , \hfill \\ \eta_{33} = \frac{{\cos^{4} \theta }}{{\alpha^{2} }},\;\eta_{44} = \frac{{ - 4\sin^{2} \theta \cos^{2} \theta \sin^{2} \phi }}{{\alpha^{2} }},\;\eta_{55} = \frac{{ - 4\sin^{2} \theta \cos^{2} \theta \cos^{2} \phi }}{{\alpha^{2} }},\;\eta_{66} = \frac{{4\sin^{4} \theta \sin^{2} \phi \cos^{2} \phi }}{{\alpha^{2} }}. \hfill \\ \end{aligned}$$

(30)

The calculation of the scattering function is given by

$$\begin{aligned} {\mathbf{S}}\left( {{\mathbf{r}}_{0} } \right) = \Delta \rho \cos 2\theta + \Delta C\eta \hfill \\ = \Delta \rho \cos 2\theta + \frac{{\sin^{4} \theta \cos^{4} \phi }}{{\alpha^{2} }}\left[ {\Delta \left( {\lambda { + }2\mu } \right) - \left( {\lambda { + }2\mu } \right)\delta_{{\Delta_{\text{N}} }} } \right] \hfill \\ + 2\left( {\frac{{\sin^{4} \theta \sin^{2} \phi \cos^{2} \phi + \sin^{2} \theta \cos^{2} \theta \cos^{2} \phi }}{{\alpha^{2} }}} \right)\left( {\Delta \lambda - \lambda \delta_{{\Delta_{\text{N}} }} } \right) \hfill \\ + \frac{{\sin^{4} \theta \sin^{4} \phi }}{{\alpha^{2} }}\left[ {\Delta \left( {\lambda { + }2\mu } \right) - \frac{{\lambda^{2} }}{{\lambda { + }2\mu }}\delta_{{\Delta_{\text{N}} }} } \right] + \frac{{\cos^{4} \theta }}{{\alpha^{2} }}\left[ {\Delta \left( {\lambda { + }2\mu } \right) - \frac{{\lambda^{2} }}{{\lambda { + }2\mu }}\delta_{{\Delta_{\text{N}} }} } \right] \hfill \\ + \frac{{2\sin^{2} \theta \cos^{2} \theta \sin^{2} \phi }}{{\alpha^{2} }}\left( {\Delta \lambda - \frac{{\lambda^{2} }}{M}\delta_{{\Delta_{\text{N}} }} } \right) - \frac{{4\sin^{2} \theta \cos^{2} \theta \sin^{2} \phi }}{{\alpha^{2} }}\Delta \mu \hfill \\ - \frac{{4\sin^{2} \theta \cos^{2} \theta \cos^{2} \phi }}{{\alpha^{2} }}\left( {\Delta \mu - \mu \delta_{{\Delta_{\text{T}} }} } \right) + \frac{{4\sin^{4} \theta \sin^{2} \phi \cos^{2} \phi }}{{\alpha^{2} }}\left( {\Delta \mu - \mu \delta_{{\Delta_{\text{T}} }} } \right) \hfill \\ \end{aligned} .$$

(31)

Substituting Eq. (31) into Eq. (4) yields

$$\begin{aligned} R_{{\text{PP}}} = \frac{1}{{4\rho \cos^{2} \theta }}\left\{ \begin{aligned} \Delta \rho \cos 2\theta + \frac{1}{{\alpha^{2} }}\Delta \left( {\rho \alpha^{2} } \right) + \left( {\frac{{ - 8\sin^{2} \theta \cos^{2} \theta }}{{\alpha^{2} }}} \right)\Delta \left( {\rho \beta^{2} } \right) \hfill \\ - \rho \delta_{{\Delta_{\text{N}} }} \left[ {2\left( {\sin^{2} \theta \sin^{2} \phi + \cos^{2} \theta } \right)g - 1} \right]^{2} \hfill \\ - \left( {4\sin^{4} \theta \sin^{2} \phi \cos^{2} \phi - 4\sin^{2} \theta \cos^{2} \theta \cos^{2} \phi } \right)\rho g\delta_{{\Delta_{\text{T}} }} \hfill \\ \end{aligned} \right\} \hfill \\ \begin{array}{*{20}c} {} \\ \end{array} = \frac{1}{{4\rho \cos^{2} \theta }}\Delta \rho \cos 2\theta + \frac{1}{{4\rho \cos^{2} \theta }}\frac{1}{{\alpha^{2} }}\left( {\alpha^{2} \Delta \rho + 2\alpha \rho \Delta \alpha } \right) \hfill \\ \begin{array}{*{20}c} {} \\ \end{array} + \frac{1}{{4\rho \cos^{2} \theta }}\left( {\frac{{ - 8\sin^{2} \theta \cos^{2} \theta }}{{\alpha^{2} }}} \right)\left( {\beta^{2} \Delta \rho + 2\beta \rho \Delta \beta } \right) - \frac{1}{{4\cos^{2} \theta }}\delta_{{\Delta_{\text{N}} }} \left[ {2\left( {\sin^{2} \theta \sin^{2} \phi + \cos^{2} \theta } \right)g - 1} \right]^{2} \hfill \\ - \frac{1}{{\cos^{2} \theta }}\left( {\sin^{4} \theta \sin^{2} \phi \cos^{2} \phi - \sin^{2} \theta \cos^{2} \theta \cos^{2} \phi } \right)g\delta_{{\Delta_{\text{T}} }} \hfill \\ \begin{array}{*{20}c} {} \\ \end{array} = \frac{1}{{2\cos^{2} \theta }}\frac{\Delta \alpha }{\alpha } - 4g\sin^{2} \theta \frac{\Delta \beta }{\beta } + \left( {\frac{1}{2} - 2g\sin^{2} \theta } \right)\frac{\Delta \rho }{\rho } \hfill \\ \begin{array}{*{20}c} {} \\ \end{array} - \frac{1}{{4\cos^{2} \theta }}\left[ {2\left( {\sin^{2} \theta \sin^{2} \phi + \cos^{2} \theta } \right)g - 1} \right]^{2} \delta_{{\Delta_{\text{N}} }} \hfill \\ \begin{array}{*{20}c} {} \\ \end{array} - \left( {\sin^{2} \theta \tan^{2} \theta \sin^{2} \phi \cos^{2} \phi - \sin^{2} \theta \cos^{2} \phi } \right)g\delta_{{\Delta_{\text{T}} }} \hfill \\ \end{aligned} .$$

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