A spectral element method for two-dimensional wave propagation in horizontally layered saturated porous media

Abstract

The general equilibrium equations describing the dynamic response of a porous saturated medium form a system of coupled hyperbolic partial differential equations. Restricting to two-dimensional plane strain wave propagation, an analytical solution for the dilatational and shear wave contributions to the displacement vectors can be found by a transformation of the generalized coordinates ( x, z, t) to ( k x, z, ω). A spectrally formulated element uses these frequency and horizontal wave number dependent eigenvectors as shape functions in a displacement formulation. The mass distribution is treated exactly without the need to subdivide an element into smaller elements and therefore, wave propagation is treated exactly. Saturated throw-off and layer elements are developed and enable—together with the dry elements as proposed by Rizzi and Doyle—the study of the harmonic and transient response of horizontally layered saturated and dry porous media. The benefits of the solution method are demonstrated by a numerical example.