Elastic Impedance Matrix-Based Reflectivity Approximation for Vertically Fractured Shales

Shale is typically studied as a transversely isotropic medium with a vertical axis of symmetry (VTI). This model can be extended to vertical fractures in transversely isotropic (VFTI) media when the shale background contains a set of vertical fractures. The reflection response of VFTI media exhibits azimuthal anisotropy with different types and strengths, which is significant for characterizing the fractures using observed data. However, shales may exhibit strong anisotropy, while fractures typically exhibit weak anisotropy, making it difficult for conventional methods to describe the reflection response of VFTI media effectively. We propose a new method for approximating the PP reflectivity of VFTI media based on the elastic impedance matrix, which effectively overcomes the strong anisotropy of the background VTI media. First, considering the weak anisotropy of the fracture, the stiffness matrix of the VFTI media is simplified using a Taylor expansion of the anisotropy parameters associated with the fracture. Then, based on the Christoffel equation, the phase velocity and polarization vector approximations for the VFTI media are derived using the exact numerical solutions corresponding to the VTI media. Finally, the elastic impedance matrix of the VFTI media is given by the stress–strain relationship and is used to construct the PP reflectivity in the weak-contrast half-spaces. The proposed PP reflectivity consists of the weak-contrast reflection of VTI media and the weakly anisotropic fracture perturbation terms. Applications to model and field data show that the proposed reflection is feasible and effective in shales containing a set of vertical fractures.