# Seismic AVOA Inversion for Weak Anisotropy Parameters and Fracture Density in a Monoclinic Medium
## 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 variationwith offset (AVO) analysis [7; 8; 9], and that soil and rock anisotropy can markedly affect the dynamic and seismic response of geostatructures [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].