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