The purpose of this paper is to derive the Green's function of an anisotropic elastic medium and validate it with the effective stiffness tensor of Barnett Shale. We have derived the frequency‐dependent Green's function by using the spectral theorem for matrices, thus simplifying the process of computing Green's function and obtaining analytical solutions. Evaluating the inverse of the Green–Christoffel tensor is an essential part of computing the Green's function. Based on the degeneracy of the eigenvalues of the Green–Christoffel tensor, the inverse of the Green–Christoffel tensor is expressed in the form of partial fractions. Consequently, the stiffness tensors from the measurement of core samples of Barnett Shale are used to validate the Green's function. We use the generalized singular approximation method of effective medium theory to model the effective stiffness of the core samples from microstructural properties. The generalized singular approximation method also allows us to compute the theoretical stiffness tensor of the Barnett Shale for porosity variations. The behaviour of the Green's function, which reflects the behaviour of the media, is studied in the static, low‐ and high‐frequency domains and under different physical parameters. It is observed that the variations of crack‐induced porosity produce different trends in Green's function for vertically transverse isotropic and horizontally transverse isotropic media. Thus, the variation of porosity is observed to be influential in differentiating between transverse isotropic media that have inclusions. The Green's function results presented in this paper have direct applications in the construction of synthetic seismograms for unbounded transversely isotropic media.

中文翻译：

---

Green函数法研究Barnett页岩的有效各向异性

本文的目的是推导各向异性弹性介质的格林函数，并使用Barnett页岩的有效刚度张量对其进行验证。我们通过使用矩阵的频谱定理来推导频率相关的格林函数，从而简化了计算格林函数并获得解析解的过程。评估Green-Christoffel张量的倒数是计算Green函数的重要部分。基于Green-Christoffel张量本征值的简并性，Green-Christoffel张量的倒数以部分分数的形式表示。因此，使用Barnett页岩岩心样品的刚度张量来验证格林函数。我们使用有效介质理论的广义奇异逼近方法从微观结构特性对岩心样品的有效刚度进行建模。广义奇异逼近方法还使我们能够计算出Barnett页岩的理论刚度张量，用于孔隙率变化。在静态，低频和高频域以及不同物理参数下研究了格林函数的行为，该行为反映了媒体的行为。可以看出，对于垂直横向各向同性和水平横向各向同性的介质，裂纹诱导孔隙率的变化在格林函数中产生不同的趋势。因此，观察到孔隙度的变化对区分具有夹杂物的横向各向同性介质有影响。绿色的'