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

Abstract

In this paper, we investigate wave propagation in orthotropic poroelastic media by studying the timedomain 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 Padé (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.

Energy analysis and discretization of the time-domain equivalent fluid model for wave propagation in rigid porous media

“…By injecting (45) and ( 46) into the EFM equations (1) and by using the original strategy brought out in [67], which consists in using the partial fraction decomposition…”

“…A diffusive representation of convolution operators leads to express them in the time domain by a continuum of diffusive variables. These variables, known as memory variables in geophysics [19] and auxiliary variables in acoustic [25,67], satisfy a first-order ordinary differential equation (ODE) easier to manage for stability analysis and numerical schemes than an explicit formula of the convolution products. Based on this approach, Blanc et al [13,14] used a diffusive representation of the shifted fractional derivative involved in the dynamic tortuosity of the JCA model.…”

“…It results in an augmented system which does not require storage of previous time-step solutions and can be solved by classical time-integration schemes. Moreover, the ADE method applied after a minor recasting of the MM expression can ease the numerical computation for schemes based on fluxes, as shown by Xie et al [67].…”

“…The real parameters are known to describe dissipative processes, while the complex conjugate parameters were recently shown to be related to locally-resonant behavior [3]. By contrast, Xie et al [67] used only real parameters to approach the dynamic tortuosity in the Biot-JKD model, a choice justified by the IRF of α. Ou [52] showed that a Padé approximation of the IRF of α can be described by real weights and poles, with a warranty of their signs. Moreover, Blanc et al [13] built the Biot-DA model with the dynamic tortuosity straightforwardly expressed as a real-parameter MM.…”

“…It brings out the main novelty of the present work by extending the wellposedness proof of the Biot-JKD model [12] for the JCAPL-EFM. Then in Section 4, a multipole-based approximation of α and β is taken and the ADE method is applied following [67,4,14]. Next, a stability analysis on the system obtained is performed, giving sufficient conditions on the multipole-based approximation to ensure the stability of the approximated EFM.…”