تحلیل دینامیکی محیط متخلخل ناهمگن اشباع شده با سیال منفذی غیرلزج با نفوذپذیری متغیر در عمق تحت بارگذاری قائم یکنواخت

The behavior of saturated porous media depends on the interaction of its various components, primarily the solid and liquid phases. This interaction is effectively described by Poroelastic theory, which integrates the mechanics of solid deformation with fluid flow within the porous structure. In the current study, the governing equations are derived from a simplified version of Biot's theory, as introduced by Zienkiewicz. This formulation employs two primary variables: the displacement of the solid phase and the pore fluid pressure. The medium under investigation is modeled as a heterogeneous porous half-space saturated with an inviscid fluid, with its permeability varying exponentially with depth. Such a configuration is subjected to axisymmetric loading conditions, making it a suitable case for analyzing the dynamic responses of porous media with spatially varying material properties. The heterogeneity of the medium with depth leads to the addition of new terms to the governing equations. These equations are derived for a heterogeneous isotropic medium with depth-dependent permeability in cylindrical coordinates, under axisymmetric conditions. The obtained mathematical equations involve coupled differential equations that are not easily solvable. In this study, a scalar potential function is introduced to solve these equations, and by applying it to the complex and coupled governing equations, a decoupled partial differential equation is obtained. Solving this equation allows for the determination of the unknowns in the problem. This partial differential equation can be solved in the frequency domain using the Hankel transform under axisymmetric conditions. The transformed equations in the Hankel-frequency domain, after being solved using the potential function, are returned to the real domain through the inverse Hankel transform. In this way, the harmonic response of the system can be computed. Due to the necessity of using the inverse Hankel transform, the responses are presented as complex, semi-infinite integrals, which require numerical methods for evaluation due to the presence of poles and branch points along the integration path. These integrals have been computed using the Method of Residues. Three distinct permeability variation indices are utilized to quantify the heterogeneity of the medium. These indices provide insights into how changes in permeability influence the mechanical and hydraulic responses of the porous medium. The results show that as permeability decreases with depth, the range of displacements in the medium becomes limited. At the excitation frequency of ω0=0.5 , the damping of responses relative to a homogeneous medium increase with distance from the loading location. At ω0=2 , the vertical displacement of both homogeneous and heterogeneous media converges approximately for z/a≥4 . Additionally, the results indicate that the location of the maximum pore fluid pressure shifts closer to the half-space surface in a heterogeneous medium compared to a homogeneous one. For instance, in a homogeneous medium, the maximum pore fluid pressure occurs at a depth of z/a=1.7 , while in a medium with a permeability parameter of δκ=-0.5 , it occurs at z/a=0.8 . This shift indicates that heterogeneity influences the depth distribution of fluid pressure, potentially affecting the medium’s overall stability and fluid flow patterns. The implications of these findings extend to various engineering and geophysical applications, including reservoir engineering, groundwater hydrology, and the design of foundations in porous media. Understanding how permeability variations affect the mechanical and hydraulic behavior of saturated porous media is crucial for predicting and mitigating risks associated with dynamic loading conditions. This study contributes to the broader understanding of poroelastic phenomena in heterogeneous systems and provides a foundation for future investigations into more complex geometries and material behaviors.