## 1. Introduction

In a variety of complex oil and gas reservoirs, fractured reservoirs widely exist in shale and igneous rock. Natural and induced fractures in reservoirs play an important role in determining fluid flow during production, and knowledge of the orientation and density of fractures is useful to optimize production from fractured reservoirs [

1]. Areas of high fracture density may represent ‘sweet spots’ of high permeability, and it is important to be able to target such locations for infill drilling [

2]. Recent developments in geophysics have revealed the viscous behavior of the underground strata, and demonstrated that the propagation of seismic waves has dispersion and attenuation, for fluid discrimination in porous rocks [

3,

4,

5]. However, there are methods available to compensate for dispersion and attenuation. In the current implementation of the method, we also ignore the loss of seismic amplitude due to attenuation. Seismic anisotropy is defined as the dependence of seismic velocity upon angle. P-waves propagating parallel to fractures are faster than those propagating perpendicular to fractures [

6]. For transversely isotropic (TI) media, empirical and analytical studies have shown that the presence of anisotropy can significantly distort conventional amplitude variation with offset (AVO) analysis [

7,

8,

9], and that soil and rock anisotropy can markedly affect the dynamic and seismic response of geostructures [

10,

11,

12].

Previous anisotropic models used to invert the seismic response of fractured reservoirs often assume a single set of aligned fractures, with horizontally transverse isotropic (HTI) symmetry [

13,

14], whereas most reservoirs contain several sets of fractures with variable orientations within a given fracture set [

15,

16,

17]. Furthermore, in the HTI model, horizontal layering of the crust, leading to a polar anisotropic background (vertical transverse isotropic, VTI), is ignored. Hence, HTI is not a suitable model for the sedimentary crust. Reflection amplitudes have advantages over seismic velocities for characterizing fractured reservoirs, because they have higher vertical resolution, and are more sensitive to the properties of the reservoir. Therefore, amplitude versus offset and azimuth (AVOA) inversion is an effective method for predicting fractures [

18]. The Zoeppritz equation for HTI anisotropy is obtained by linearized approximation [

13]. Using the linear P-wave reflection coefficient formula, the feasibility of fracture compliance tensor inversion in orthorhombic and monoclinic media is analyzed [

6,

19]. In the inversion process, it is assumed that the weak anisotropy (WA) parameters of VTI background media are known. Under the assumption of HTI and orthogonal media, the elastic impedance (EI) is re-expressed in terms of the normal and tangential fracture weaknesses, and the EI inversion for fracture weaknesses is implemented, based on the linear-slip model [

20,

21]. Based on HTI media, many scholars have used normal move out (NMO) velocity elliptical inversion, or prestack AVOA inversion, to indirectly calculate fracture density. Scholars [

22] have implemented a qualitative analysis of fracture density and orientation, based on P-wave (PP) and converted wave (PS) velocity anisotropy. They have used elliptical inversion to estimate the magnitude of anisotropy associated with fracture density, from PP data, and then qualitatively estimated fracture density, from time differences of the fast and slow PS velocities. It has been suggested that the difference in NMO times be used to estimate the axial ratio of the ellipse, which is proportional to fracture density [

23]. A non-linear Bayesian inversion of the seismic amplitude versus incident and azimuthal angle (AVAZ) and dynamic production data is used to calculate the fracture density and aperture for an HTI fractured reservoir in which the fracture parameters are invariant [

24]. The prestack AVAZ inversion method is used to extract the anisotropy parameters or shear-wave splitting factor,

$\gamma $, which is related to fracture density [

25]. The inversion algorithm has also been deeply studied by many scholars [

26,

27,

28].

However, with indirect substitution there exists intrinsic errors, and the inversion methods based on simplified models such as the HTI medium can be misleading when applied to shale fractured reservoirs with a complex medium. These limit the improvement to the prestack inversion accuracy. Therefore, it cannot satisfy the accuracy requirements for obtaining fracture density.

In this paper, based upon previous research, and combined with rock physics theories, assuming that the fracture reservoir has monoclinic symmetry, and under the condition that the fracture radius, width, and orientation are known, we derived a new form of P-wave reflection coefficient, consisting of the WA parameters, and the fracture density. This was achieved by substituting the stiffness coefficients of the VTI background, and normal and tangential fracture compliances. The new form of P-wave reflection coefficient was then used to build a theoretical framework of linear AVOA inversion, based on Bayesian theory, that can avoid the assumption that the WA parameters of VTI background media are known, and avoid the intrinsic errors introduced from the indirect substitution. Finally, synthetic data, which contain Gaussian random noise with different signal-to-noise ratios (SNRs), were utilized to verify the stability of the proposed approach.

## 3. Numerical Analysis

Synthetic seismic data were created by using interpretation results of well log data and vertical seismic profiling (VSP) data, which provided input for the AVOA inversion. It was assumed that the fracture medium had monoclinic symmetry, the azimuths of the two fracture sets were 10° and 70°, respectively, the average fracture width was 1 mm, the average fracture radius was 10 m, and the average area of the fracture was 0.02 m

^{2}.

Figure 2 shows two sets of fracture densities from the well log interpretation results as model parameters for inversion.

Figure 3 shows the well log and VSP interpretation results, among which the isotropic elastic parameters were used as the known parameters, to calculate the reflection coefficient of isotropic background and sensitivities.

In order to test the robustness of the proposed inversion algorithm, synthetic seismic data were created by the model parameters shown in

Figure 2 and

Figure 3, as well as a 30 Hz Ricker wavelet, based on the derived PP-wave reflection coefficient, Equation (13), and the convolution model shown in

Figure 4a. The incident angle had 10 angles between 0° and 45°, and the azimuths were 10°, 55°, 100°, and 145°.

Gaussian random noises (the signal-to-noise ratios (SNRs) were 5 and 2) were added into the synthetic data, as shown in

Figure 4b,c. Comparisons between the inversion results (black) of the WA parameters, the fracture density, and the true values (red) are displayed in

Figure 5. We can see that, in the case of the SNR being 5, the proposed inversion method could make a stable estimation of WA parameters and fracture density; however, when the SNR was 2, the inversion results of fracture density exhibited more instabilities. Random noise had some influence on the WA parameters and fracture density, but the overall agreement was good. In general, it can be concluded that the proposed inversion method is stable and valid.

In the approach proposed, the fracture width, radius, and strike are required to be assumed. The dimensions of fractures are a priori functions of many factors in the medium and cannot be readily assumed in practice. In order to test the impact of this assumption (the dimensions assumed for the fractures) on the predicted results, the results of inversion, using the same synthetic data as used in

Figure 4a, but with the forward operators, that fracture radius is 0.25 m and fracture width is 5 mm (wrong assumptions), are shown in

Figure 6a,b, respectively. Comparing the inversion results with

Figure 5a, we observed that, in the case of the fracture radius being 0.25 m, the incorrect assumption of fracture radius introduced larger errors into the prediction of WA parameters and fracture density. In general, the inverted model parameters captured the amplitudes and variability of the true models. When the fracture width was 5 mm, the incorrect assumption of fracture width almost had no impact on WA parameters; however, the accuracy and stability of the fracture density inversion was seriously influenced. More accurate information on fracture radius and width may help to improve the accuracy of WA parameters and fracture density inversion.

In order to demonstrate the advantage of the proposed algorithm in fracture density estimation, we made use of the same synthetic data as used in the

Figure 4a and estimated anisotropy through elliptical inversion of interval velocities. The resulting NMO velocity varied ellipsoidally with the azimuth. The anisotropy intensity was defined by the axial ratio. To compare with the true value of fracture density, the anisotropy intensity, multiplied by a factor, is shown in

Figure 7. We can see that the accuracy of the proposed inversion, in relative terms, is greater than that of elliptical inversion.

## 4. Conclusions

Assuming that the fractured reservoir has a monoclinic symmetry of VTI background, a new form of P-wave reflection coefficient, in terms of WA parameters and fracture density, was obtained by substituting the stiffness matrix coefficient of the VTI background, and normal and tangential fracture compliances. The new expression of the reflection coefficient for a monoclinic medium can avoid calculation errors due to the assumption of a simple model (such as an HTI medium). Additionally, the inversion parameters contain WA parameters, which avoid the assumption that the parameters of the VTI background are known in the traditional form of a P-wave reflection coefficient. Finally, we achieved the direct inversion of WA parameters and fracture density, and avoided the intrinsic errors introduced by indirect substitution. The synthetic test demonstrated that this method can be used to accurately estimate WA parameters and fracture density, which significantly benefits further calculation of reservoir porosity or permeability, the prediction of the “sweet spot”, and the evaluation of a reservoir’s reconstruction.

In the approach proposed, the fracture width, radius, and strike are required to be assumed. Incorrect assumptions of fracture width and radius will yield erroneous results, for WA parameters and fracture density, in practice. A priori knowledge about fracture width is important in the prediction of fracture density. However, the interpretation of fracture reservoirs shows that the fracture apertures in reservoir depth are about 1 mm, and have little variation.