# Azimuthal elastic impedance-based Fourier coefficient variation with angle inversion for fracture weakness

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## Abstract

Quantitative inversion of fracture weakness plays an important role in fracture prediction. Considering reservoirs with a set of vertical fractures as horizontal transversely isotropic media, the logarithmic normalized azimuthal elastic impedance (EI) is rewritten in terms of Fourier coefficients (FCs), the 90° ambiguity in the azimuth estimation of the symmetry axis is resolved by judging the sign of the second FC, and we choose the FCs with the highest sensitivity to fracture weakness and present a feasible inversion workflow for fracture weakness, which involves: (1) the inversion for azimuthal EI datasets from observed azimuthal angle gathers; (2) the prediction for the second FCs and azimuth of the symmetry axis from the estimated azimuthal EI datasets; and (3) the estimation of fracture weakness combining the extracted second FCs and azimuth of the symmetry axis iteratively, which is constrained utilizing the Cauchy sparse regularization and the low-frequency regularization in a Bayesian framework. Tests on synthetic and field data demonstrate that the 90° ambiguity in the azimuth estimation of the symmetry axis has been removed, and reliable fracture weakness can be obtained when the estimated azimuth of the symmetry axis deviates less than 30°, which can guide the prediction of fractured reservoirs.

## Keywords

Fracture weakness Azimuth of the symmetry axis Azimuthal Fourier coefficients HTI Azimuthal elastic impedance## 1 Introduction

Subsurface fractures contribute to providing pathways for fluid flow and increasing the permeability of reservoirs. The key parameters of great interest for fractured reservoir exploration are the distribution and orientation of fracture systems, which play an important role in optimizing production from naturally fractured reservoirs (Sayers 2009; Far et al. 2014; Den Boer and Sayers 2018).

The seismic fracture prediction can be classified into two basic types: One is the seismic fracture qualitative prediction, which is mainly based on the discontinuity of fractures and uses the prestack seismic diffraction wave imaging approach or poststack geometric seismic attributes (such as curvature, coherence, and discontinuity) to qualitatively describe macroscale fractures, such as faults. The other is the seismic fracture quantitative prediction, which is mainly based on seismic azimuthal anisotropy (such as reflection amplitude, velocity, and attenuation) to predict mesoscale fractures, including obtaining fracture density and direction by ellipse fitting using azimuthal anisotropy attributes and quantitatively inverting fracture parameters using azimuthal prestack seismic data (Liu et al. 2015; Chen et al. 2016; Liu et al. 2018; Yuan et al. 2019).

Fracture detection utilizing seismic anisotropy has been an active field in the past decades. Hydrocarbon reservoirs with vertically aligned fractures can be described by horizontally transverse isotropic (HTI) media caused by a single set of rotationally invariant fractures in isotropic background rock. The amplitude and velocity of seismic waves propagating in HTI media usually exhibit significant azimuthal anisotropy. Reflection amplitude is superior to seismic velocities in characterizing fractured reservoirs due to its higher vertical resolution and sensitivity to the properties of a reservoir (Far et al. 2013). Much work has been done in deriving linear P-wave reflection coefficients in HTI media (Rüger 1998; Pšencík and Martins 2001; Shaw and Sen 2006; Chen et al. 2014a; Pan et al. 2018a) and directly characterizing fractured zones using azimuthal P-wave reflectivity data (Mallick et al. 1998; Gray and Todorovic-Marinic 2004; Bachrach et al. 2009; Mahmoudian et al. 2015; Wang et al. 2019).

In recent years, amplitude variation with angle and azimuth (AVAZ) inversion combining with the linear slip theory (Schoenberg and Douma 1988; Schoenberg and Sayers 1995) used for modeling fractures embedded in isotropic host rock is widely applied to predict fractured reservoirs. Based on the rock physics model, Chen et al. (2014a) proposed the AVAZ inversion method to estimate elastic and fracture weakness parameters. Pan et al. (2019) presented Bayesian AVAZ direct inversion for fluid indicator and fracture weakness in an oil-bearing fractured reservoir. AVAZ inversion is an ill condition; a large number of unknown parameters and the cross talk between the elastic and fracture parameters inevitably make several-term AVAZ inversion unstable (Downton et al. 2006; Bachrach 2015). One tentative approach to reduce the number of unknown parameters is to implement azimuthal seismic amplitude difference inversion for fracture weakness estimation (Chen et al. 2017a; Pan et al. 2017; Xue et al. 2017). In addition, Downton and Roure (2011, 2015) rewrote azimuthal P-wave reflectivity in terms of Fourier coefficients (FCs) and proposed an azimuthal Fourier coefficient elastic inversion for fracture parameters estimation. Barone and Sen (2018) implemented the application of real seismic data targeting the Haynesville Shale using a Fourier azimuthal amplitude variation fracture characterization method. Nevertheless, the suitability and reliability of AVAZ inversion in the presence of noise are still controversial. Moreover, it is a challenging task to extract reasonable space variant wavelets of each incident angle at different azimuths used for AVAZ inversion. This can be addressed by azimuthal elastic impedance (EI) extended from the concept of EI (Connolly 1999; Whitcombe 2002; Wang et al. 2006; Yin et al. 2013; Zong et al. 2013, 2016; Mozayan et al. 2018). In the last decades, there were abundant studies on the application of azimuthal EI in anisotropic inversion (Martins 2006; Chen et al. 2014b, 2017b; Pan et al. 2018b; Pan and Zhang 2019).

Our study is the extension of Downton and Roure (2015) research, which stabilizes the inversion process by treating the fracture weakness separately from the elastic parameters using FCs. Different from traditional fracture weakness inversion using azimuthal reflection amplitude or azimuthal EI or azimuthal reflection amplitude FCs, in this paper, a novel azimuthal EI-based FC variation with angle inversion method is proposed. The normalized azimuthal EI equation in HTI media is rewritten by using the Fourier series expansion method, the azimuth of the symmetry axis is estimated without 90° ambiguity using FCs, and FCs with the highest sensitivity to fracture weakness are chosen to estimate fracture weakness, which is constrained utilizing the Cauchy sparse regularization and the low-frequency regularization in a Bayesian framework. The influence of the azimuth of the symmetry axis on fracture weakness inversion is analyzed, and the proposed approach is demonstrated on both synthetic and real data.

## 2 Method and theory

### 2.1 Fourier coefficients of normalized azimuthal EI in logarithm domain

^{–}represents the average of elastic parameters. \(g = {{\overline{\beta }^{2} } \mathord{\left/ {\vphantom {{\overline{\beta }^{2} } {\overline{\alpha }^{2} }}} \right. \kern-0pt} {\overline{\alpha }^{2} }}\) is the ratio of the squared vertical S- and P-wave velocities in the background isotropic medium, and \(\Delta_{\text{N}}\) and \(\Delta_{\text{T}}\) are the normal and tangential fracture weakness of layers. If the HTI model is induced by penny-shaped cracks, \(\Delta_{\text{T}}\) gives a direct estimate of the crack density and the ratio \({{\Delta_{\text{N}} } \mathord{\left/ {\vphantom {{\Delta_{\text{N}} } {\Delta_{\text{T}} }}} \right. \kern-0pt} {\Delta_{\text{T}} }}\) is a sensitive indicator of fluid saturation (Bakulin et al. 2000). \(\Delta_{{\Delta_{\text{N}} }}\) represents the difference in normal weakness between the upper and lower layers, and \(\Delta_{{\Delta_{\text{T}} }}\) represents the difference in tangential weakness between the upper and lower layers. \(\theta\) is the incident angle, and \(\varphi = \phi - \phi_{\text{sym}}\) is the angle between the observed azimuth \(\phi\) and azimuth \(\phi_{\text{sym}}\) of the symmetry axis.

Equation (3) is the basis of conventional EI variation with angle and azimuth inversion for elastic and fracture weakness parameters. However, the robustness of inverting fracture weakness from azimuthal EI is still controversial due to the coupling between elastic and fracture weakness parameters.

*n*= 0, 2, 4) represent the nth FCs, respectively. For the case of

*X*regularly sampled data, a discrete Fourier transform (DFT) can be used to calculate \(a_{n} \left( \theta \right)\) and \(b_{n} \left( \theta \right)\):

Note that Nyquist criterion must be met so that \(a_{n} \left( \theta \right)\) and \(b_{n} \left( \theta \right)\) can be obtained by DFT. In order to estimate \(a_{ 2} \left( \theta \right)\) and \(b_{ 2} \left( \theta \right)\), the data must be sampled no coarser than 45**°**. In order to estimate \(a_{ 4} \left( \theta \right)\) and \(b_{ 4} \left( \theta \right)\), the data must be sampled no coarser than 22.5**°**.

### 2.2 Sensitivity analysis of the FCs

^{3}, respectively. The normal and tangential weaknesses increase from 0 to 0.2, respectively, in increments of 0.05. We use Eqs. (6) and (7) to calculate the second and fourth FCs. Figure 1a, b shows the second FC \(r_{ 2} \left( \theta \right)\) variation with the normal and tangential fracture weaknesses. Figure 2a, b shows the fourth FC \(r_{4} \left( \theta \right)\) variation with the normal and tangential fracture weaknesses. It can be seen that when the incident angle is less than 30

**°**, \(r_{4} \left( \theta \right)\) hardly changes with normal and tangential weaknesses, and \(r_{ 2} \left( \theta \right)\) changes significantly with normal and tangential weaknesses. Compared with \(r_{4} \left( \theta \right)\), \(r_{ 2} \left( \theta \right)\) is more sensitive to normal and tangential weaknesses.

### 2.3 Resolving the 90° ambiguity in the azimuth estimation of the symmetry axis

**°**ambiguity in \(\phi_{\text{sym}}\) estimate using \(a_{ 2} \left( \theta \right)\) and \(b_{ 2} \left( \theta \right)\), and there is a 45

**°**ambiguity in \(\phi_{\text{sym}}\) estimate using \(a_{4} \left( \theta \right)\) and \(b_{4} \left( \theta \right)\); we propose an approach to resolve the ambiguity. Firstly, \(\phi_{\text{sym}}\) is directly calculated from \(a_{ 2} \left( \theta \right)\) and \(b_{ 2} \left( \theta \right)\) by:

Since \(a_{ 2} \left( \theta \right) = r_{ 2} \left( \theta \right)\cos 2\phi_{\text{sym}}\), \(b_{ 2} \left( \theta \right) = r_{ 2} \left( \theta \right)\sin 2\phi_{\text{sym}}\), for the case of \(r_{ 2} \left( \theta \right) > 0\), if the sign of \(a_{ 2} \left( \theta \right)\) is the same as the sign of \(\cos 2\phi_{\text{sym}}\), and the sign of \(b_{ 2} \left( \theta \right)\) is the same as the sign of \(\sin 2\phi_{\text{sym}}\), the estimated \(\phi_{\text{sym}}\) is correct, otherwise the estimated \(\phi_{\text{sym}} = \phi_{\text{sym}} + 90^\circ\). For the case of \(r_{ 2} \left( \theta \right) < 0\), if the sign of \(a_{ 2} \left( \theta \right)\) is opposite to the sign of \(\cos 2\phi_{\text{sym}}\), and the sign of \(b_{ 2} \left( \theta \right)\) is opposite to the sign of \(\sin 2\phi_{\text{sym}}\), the estimated \(\phi_{\text{sym}}\) is correct, otherwise the estimated \(\phi_{\text{sym}} = \phi_{\text{sym}} + 90^\circ\).

Therefore, for fluid-filled cracks, there is always \(r_{ 2} \left( \theta \right) > 0\).

To judge the sign of \(r_{ 2} \left( \theta \right)\) using the above method requires knowing fracture infill (content) in advance, but fracture infill is usually unknown, so anisotropic rock physical model (Zhang et al. 2013; Chen et al. 2014a; Pan et al. 2017) can be used to estimate fracture weaknesses, then calculate \(r_{ 2} \left( \theta \right)\), and judge its sign to further resolve ambiguity.

### 2.4 Azimuthal EI-based FC variation with angle inversion for fracture weakness

*M*incident angle and

*N*time sample, combining Eqs. (10) and (11), we can get the following matrix expression:

Here, the superscript \({\text{T}}\) denotes the transpose of a matrix, and the symbol \(\text{diag}\) denotes a diagonal matrix; the subscripts *m* represents the *m*th angle of incidence.

### 2.5 Workflow of azimuthal EI-based FC variation with angle inversion for fracture weakness

- 1.
The constrained sparse spike inversion (CSSI) for azimuthal EI datasets from azimuthal angle stacks. The inputs include azimuthal angle stacks, azimuthal seismic wavelets, and low-frequency azimuthal EI models of each incident angle. Since azimuth \(\phi_{\text{sym}}\) of the symmetry axis is unknown, low-frequency azimuthal EI models are obtained by interpolating isotropic EI calculated from well log data.

- 2.
The prediction for \(a_{ 2} \left( \theta \right)\) and \(b_{ 2} \left( \theta \right)\) from all estimated azimuthal EI datasets and estimation for \(\phi_{\text{sym}}\) from \(a_{ 2} \left( \theta \right)\) and \(b_{ 2} \left( \theta \right)\). The ambiguity in \(\phi_{\text{sym}}\) estimate is resolved by judging the sign of \(r_{ 2} \left( \theta \right)\), which is obtained by combining fracture weaknesses from the anisotropic rock physics model (Zhang et al. 2013; Chen et al. 2014a; Pan et al. 2017) and elastic parameters from well log. Note that for the case of regularly sampled azimuthal EI datasets, \(a_{ 2} \left( \theta \right)\) and \(b_{ 2} \left( \theta \right)\) can be calculated using a DFT shown in Eqs. (14) and (15). For the more complex case of irregularly sampled EI datasets in azimuth, a least squares inversion may be performed by using Eq. (8).

- 3.
The estimation for fracture weaknesses using the extracted \(a_{ 2} \left( \theta \right)\), \(b_{ 2} \left( \theta \right)\), and \(\phi_{\text{sym}}\) iteratively, which is constrained utilizing the Cauchy sparse regularization and the low-frequency regularization in a Bayesian framework (Pan and Zhang 2019).

## 3 Examples

### 3.1 Synthetic examples

Elastic and anisotropic parameters of the five-layer model (Rüger and Tsvankin 1997)

Layers | \(\alpha ,{\text{ km/s}}\) | \(\beta ,{\text{ km/s}}\) | \(\rho ,{\text{ g/cm}}^{ 3}\) | \(\varepsilon^{\left( V \right)}\) | \(\delta^{\left( V \right)}\) | \(\gamma\) | \(\phi_{\text{sym}} , \, ^\circ\) |
---|---|---|---|---|---|---|---|

Isotropic | 4.500 | 2.530 | 2.800 | 0.000 | 0.000 | 0.000 | – |

HTI | 4.388 | 2.530 | 2.800 | − 0.150 | − 0.155 | 0.085 | 60 |

Isotropic | 4.500 | 2.530 | 2.800 | 0.000 | 0.000 | 0.000 | – |

HTI | 4.388 | 2.530 | 2.800 | − 0.150 | − 0.155 | 0.085 | 120 |

Isotropic | 4.500 | 2.530 | 2.800 | 0.000 | 0.000 | 0.000 | – |

### 3.2 Field data example

## 4 Conclusions

- 1.
The approach based on reflection amplitude for estimating azimuth of the symmetry axis has a 90

**°**ambiguity. In this paper, the ambiguity is eliminated by judging the sign of the second FC \(r_{ 2} \left( \theta \right)\). If fracture infill is known, for example, for fluid-filled cracks, there is always \(r_{ 2} \left( \theta \right) > 0\). If the cracks are dry (gas-filled), in this case it becomes complicated to determine the sign of \(r_{ 2} \left( \theta \right)\), which is related to incident angle and the ratio \(g\). For the case of \(\theta \le 30^\circ\), if \(g > 0.4\), there is always \(r_{ 2} \left( \theta \right) > 0\). For the case of \(\theta > 30^\circ\), if \(g < 0.4\), there is always \(r_{ 2} \left( \theta \right) < 0\). While fracture infill is usually unknown, anisotropic rock physical model can be used to estimate fracture weakness, then calculate \(r_{ 2} \left( \theta \right)\), and judge its sign to further resolve ambiguity. In addition, the analysis shows that seismic data at middle and large incident angles should be used to estimate \(\phi_{\text{sym}}\). In the case that there are multiple \(\phi_{\text{sym}}\) estimated at middle and large incident angles, the average of multiple \(\phi_{\text{sym}}\) can be a more stable and reliable estimate, which contributes to reducing the uncertainty of the \(\phi_{\text{sym}}\) estimate. - 2.
The sensitivity analysis of the FCs to fracture weaknesses shows that the second FC is more sensitive to fracture weaknesses than the fourth FC; the second FCs obtained from azimuthal EI data can be simultaneously combined to estimate fracture weaknesses. However, azimuth of the symmetry axis should be estimated prior to the fracture weakness inversion, and the fracture weakness inversion is affected by the estimated azimuth of the symmetry axis. Tests on the synthetic and real data show that the inversion results of fracture weaknesses obtained by the proposed approach are stable and reliable when azimuth of the symmetry axis deviates less than 30

**°**. - 3.
The proposed approach for estimating azimuth of the symmetry axis and fracture weaknesses depends on the constructed anisotropic rock physical model. The correctness of the model affects the accuracy of the azimuth of the symmetry axis and fracture weakness estimation, so questions remain about the reliability of this inversion, and more work needs to be done to improve the reliability of fracture prediction.

## Notes

### Acknowledgements

We thank Dr. Huaizhen Chen from CREWES, University of Calgary, Alberta, Canada, for his helpful suggestions. We would like to express our gratitude for the sponsorship of the National Natural Science Foundation of China (41674130) and National Grand Project for Science and Technology (2016ZX05002-005) for funding this research.

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