Fundamental solutions to the transversely isotropic poroelastodynamics Mandel's problem

Abstract

AbstractMany geomaterials, such as shale, and biomaterials, including human bone and cartilage, are anisotropic. Understanding the dynamic responses of such anisotropic porous media is essential in field operations and laboratory measurements. This paper presents analytical solutions to the transversely isotropic poroelastodynamics Mandel's problem, using the Fourier transform method and Cardano formula to solve coupled six‐order partial differential equations. Potential applications of the solutions include explaining coupled fluid‐solid dynamics, validating numerical algorithms, and interpreting dynamic responses of anisotropic porous media. As an example, we apply the solutions to simulate a fluid‐saturated transversely isotropic Trafalgar shale specimen subjected to harmonic excitation. The simulation shows that pore pressure builds up as frequency increases. The buildup occurs at a lower frequency for a lower horizontal permeability. The amount of pore pressure buildup due to the Mandel‐Cryer effect is sensitive to material anisotropy. The porous material's effective stiffness reaches its periodic minimum and maximum at the resonant and anti‐resonant frequencies controlled by mechanical anisotropy and pore fluid.