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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.

Stress, rock microstructure and frequency are the three key factors that influence the velocities of elastic waves and, hence, are sensitive to the Biot effective stress parameter (α) in porous rocks. The effective stress in an isotropic poroelastic medium relates to applied pressure and pore pressure, with the Biot parameter (α) as a scaling factor of the pore pressure. This paper provides an independent derivation of the tensor characteristics of α through elastic moduli, a microscopic effective medium derivation, and frequency‐dependent behaviour of α for an anisotropic medium. We provide an explicit expression, especially for isotropic rock under uniaxial stress, considering the nonlinear part of elastic constants. In the effective medium derivation, we assumed that the rock contained both isolated pores and connected pores saturated with liquid. To support our theoretical formulation, we calculated the Biot tensor of sandstone and shale by inverting the ultrasonic velocities of transversely isotropic rock under uniaxial stress where mineralogical composition and porosity are known. Even though porosity and rock microstructure play significant roles in α as stress varies, we also see as much as a 21% difference between horizontal and vertical components of α for rocks with transversely isotropic symmetry. We then estimated the frequency‐dependent Biot tensor for transversely isotropic models using numerical calculations. We noticed significant differences between vertical (α33) and horizontal (α11) components of α, especially at the surface seismic frequency band. However, uniaxial stress and horizontally aligned microstructure influence the elastic moduli and Biot tensor contrarily. In general, anisotropy due to uniaxial stress shows lower α33 and higher α11. The proposed method shows an excellent prediction of α33 and α11 for given data of uniaxial stress and vice versa.

Summary The productive zone of the Marcellus Shale has variations in rock properties such as clay content, kerogen content and aspect ratio of structural kerogen. These variations may also cause elastic anisotropy. We studied the seismic response of the Marcellus Shale for three types of models with variations in a specific parameter such as kerogen content, aspect ratio and anisotropic parameters. The reservoir was assumed transversely isotropic with vertical axis of symmetry (VTI). The seismograms were numerically simulated by applying a vertical impulse source and three component receivers. For kerogen content models, scenarios for 5%, 10%, 20% and 30% of kerogen in the host matrix were considered. For both kerogen content and aspect ratio models, PP amplitudes were most sensitive at zero to near offset while PS and SS amplitudes were most sensitive at mid offsets (12 to 30 degree angle of incidences).The anisotropic effect to angular amplitude variations for PP, PS and SS reflections were found to be dominantly controlled by the Thomsen Ɛ parameter. The anisotropic effect to PP data can be seen at large offsets (>15 o incidence angle). The most sensitive portion of PS reflections was observed at mid offsets (15 o -30 o ). The findings from these synthetic data may ultimately be applied to real data for the improved characterization of shale gas reservoirs.