Biot-JKD model: simulation of 1D transient poroelastic waves with fractional derivatives

Emilie Blanc (1) , Guillaume Chiavassa (2) , Bruno Lombard (1)

1 LMA - Laboratoire de Mécanique et d'Acoustique [Marseille]
2 ECM - École Centrale de Marseille

A time-domain numerical modeling of Biot poroelastic waves is presented. The viscous dissipation occurring in the pores is described using the dynamic permeability model developed by Johnson-Koplik-Dashen (JKD). Some of the coefficients in the Biot-JKD model are proportional to the square root of the frequency: in the time-domain, these coefficients introduce order 1/2 shifted fractional derivatives involving a convolution product. Based on a diffusive representation, the convolution kernel is replaced by a finite number of memory variables that satisfy local-in-time ordinary differential equations. Thanks to the dispersion relation, the coefficients in the diffusive representation are obtained by performing an optimization procedure in the frequency range of interest. A splitting strategy is then applied numerically: the propagative part of Biot-JKD equations is discretized using a fourth-order ADER scheme on a Cartesian grid, whereas the diffusive part is solved exactly. Comparisons with analytical solutions show the efficiency and the accuracy of this approach.

Mots clés: porous media; elastic waves; Biot-JKD model; fractional derivatives; time splitting; finite difference methods; Cartesian grid

Domaines: Acoustique [physics.class-ph]; Acoustique [physics.class-ph]; Analyse numérique [math.NA]

Emilie Blanc, Guillaume Chiavassa, Bruno Lombard. Biot-JKD model: simulation of 1D transient poroelastic waves with fractional derivatives. 2012. ⟨hal-00713127v1⟩