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.

中文翻译：

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弱正交性介质中水平和垂直裂缝特征的方位傅里叶系数反演

在各向同性背景下渗透的水平和垂直排列的裂缝（节理）会产生长波等效斜方介质，这对于天然裂缝储层是很常见的。我们已经开发出一种可行的方法，利用观测到的方位角地震反射数据来表征水平和垂直裂缝。对于弱各向异性，我们假设水平裂缝和垂直裂缝是解耦的。使用线性滑移正交各向异性模型，我们首先获得有效正交各向异性各向异性弹性介质的有效弹性刚度矩阵和相应的受扰矩阵的解析表达式。结合正交函数介质的散射函数和摄动矩阵，然后，我们根据压缩波（P波）和剪切波（S波）的模量，密度以及水平和垂直裂缝引起的法向和切向，得出有效正交斜介质的线性化PP波反射系数弱点。为了处理正交各向异性介质中多个模型参数的反问题，我们将线性化的PP波反射系数重新表达为傅立叶级数。根据傅立叶系数（FCs）对各向同性背景弹性参数和四个断裂弱点的敏感性分析，我们最终建立了一种描述正交各向异性模型的三步反演方法，该方法包括（1）从不同的入射角估计FCs。观测到的方位角地震数据和对称轴方位角的确定，（2）零阶FC的各向同性背景弹性参数和水平裂缝的断裂弱点的迭代贝叶斯反演，以及（3）垂直FC的二阶FC的裂缝的弱点的迭代贝叶斯反演。通过对合成和现场数据集的测试验证了该方法的有效性，该方法证明了P波和S波模量，密度和四个裂缝弱点的反演结果对于斜方对称对称的含气裂缝性储层表征是可靠且合理的。