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

Abstract

Poroelastic materials play an important role in biomechanical and geophysical research. In this paper, we investigate wave propagation in orthotropic poroelastic media by studying the time-domain poroelastic wave equations. Both the low frequency Biot's (LF-Biot) equations and the Biot-Johnson-Koplik-Dashen (Biot-JKD) model 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. In the Biot-JKD model, the convolution operator involves order 1/2 shifted fractional derivatives in the time domain, which is challenging to discretize.In this work, a new method of the multipoint Padé (or Rational) approximation for Stieltjes function is applied to approximate the JKD dynamic tortuosity and then an augmented system of Biot-JKD model is obtained, where the kernel of the memory term is replaced by the finite auxiliary variables satisfying a local system of ordinary differential equations. The Runge-Kutta discontinuous Galerkin (RKDG) method with the un-splitting method is used to compute the numerical solution, and numerical examples are presented to demonstrate the high order accuracy and stability of the method. Compared with the existing approaches for solving the Biot-JKD equations, the augmented system presented here require neither the storage of solution history nor the computation of the flux of the auxiliary variables.