Analytical 3D fundamental solutions for dynamic saturated poroelasticity

Abstract

The fundamental solution is a particular solution of the inhomogeneous equation with Dirac delta function as the right hand side term. It holds significant importance in both applied and theoretical mathematics and physics. This study focuses on deriving 3D fundamental solutions in the frequency domain for wave propagation in a fluid-saturated porous medium in the context of Biot's theory. The approach begins with the Helmholtz decomposition, which decomposes the variables into three scalar potentials. These potentials are then decoupled and determined via matrix eigenvalue analysis. Based on the derived potentials, the fundamental solutions for displacements and pressure are derived. Finally, the applicability of the fundamental solutions is verified via 3D cases through the method of fundamental solutions.