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In this paper, we investigate wave propagation in orthotropic poroelasticmedia by studying the time-domain poroelastic equations. Both the low frequencyBiot's (LF-Biot) equations and the Biot-Johnson-Koplik-Dashen (Biot-JKD) modelsare considered. In LF-Biot equations, the dissipation terms are proportional tothe relative velocity between the fluid and the solid by a constant. Contrastto this, the dissipation terms in the Biot-JKD model are in the form of timeconvolution (memory) as a result of the frequency-dependence of fluid-solidinteraction at the underlying microscopic scale in the frequency domain. Thedynamic tortuosity and permeability described by Darcy's law are two crucialfactors in this problem, and highly linked to the viscous force. In the Biotmodel, the key difficulty is to handle the viscous term when the pore fluid isviscous flow. In the Biot-JKD dynamic permeability model, the convolutionoperator involves order $1/2$ shifted fractional derivatives in the timedomain, which is challenging to discretize. In this work, we utilize themultipoint Pad$\acute{e}$ (or Rational) approximation for Stieltjes function toapproximate the dynamic tortuosity and then obtain an augmented system ofequations which avoids storing the solutions of the past time. The Runge-Kuttadiscontinuous Galerkin (RKDG) method is used to compute the numerical solution,and numerical examples are presented to demonstrate the high order accuracy andstability of the method.

A discontinuous Galerkin method for wave propagation in orthotropic poroelastic media with memory terms