Azimuthal Fourier coefficient inversion for horizontal and vertical fracture characterization in weakly orthorhombic media

Abstract

Horizontally and vertically aligned fractures (joints) permeated in an isotropic background can give rise to a long-wavelength equivalent orthorhombic medium, which is common for naturally fractured reservoirs. We have developed a feasible approach to characterize the horizontal and vertical fractures using observed azimuthal seismic reflection data. For weak anisotropy, we assume that the horizontal and vertical fractures are decoupled. Using a linear-slip orthorhombic model, we first obtain analytic expressions for the effective elastic stiffness matrix and the corresponding perturbed matrix of an effective orthorhombic anisotropic elastic medium. Combining the scattering function and the perturbed matrix of the orthorhombic medium, we then derive a linearized PP-wave reflection coefficient of effective orthorhombic medium in terms of compressional-wave (P-wave) and shear-wave (S-wave) moduli, density, and horizontal- and vertical-fracture-induced normal and tangential weaknesses. To handle the inverse problem for multiple model parameters in an orthorhombic medium, we reexpress the linearized PP-wave reflection coefficient as a Fourier series. According to the sensitivity analysis of Fourier coefficients (FCs) to isotropic background elastic parameters and four fracture weaknesses, we finally establish a three-step inversion approach to describe the orthorhombic model, which involves (1) estimation of the FCs at different incidence angles from observed azimuthal seismic data and determination of the symmetry axis azimuth, (2) an iterative Bayesian inversion for isotropic background elastic parameters and fracture weaknesses of horizontal fractures from the zeroth-order FC, and (3) an iterative Bayesian inversion for fracture weaknesses of vertical fractures from the second-order FC. The proposed approach is validated by tests on synthetic and field data sets, which demonstrates that the inversion results of P- and S-wave moduli, density, and four fracture weaknesses are robust and reasonable for gas-bearing fractured reservoir characterization with orthorhombic symmetry.