A time-domain numerical method for Biot-JKD poroelastic waves in 2D heterogeneous media

TL;DR

This paper introduces an explicit finite-difference method for simulating 2D Biot-JKD poroelastic waves in heterogeneous media, effectively handling fractional derivatives via diffusive representations and advanced numerical schemes.

Contribution

It develops a novel numerical approach combining diffusive representations and high-order schemes to efficiently solve Biot-JKD equations in complex media.

Findings

Accurate simulation of poroelastic wave propagation in heterogeneous media.

Effective handling of fractional derivatives without extensive memory storage.

Validation through realistic numerical experiments.

Abstract

An explicit finite-difference scheme is presented for solving the two-dimensional Biot equations of poroelasticity across the full range of frequencies. The key difficulty is to discretize the Johnson-Koplik-Dashen (JKD) model which describes the viscous dissipations in the pores. Indeed, the time-domain version of Biot-JKD model involves order 1/2 shifted fractional derivatives which amounts to a time convolution product. To avoid storing the past values of the solution, a diffusive representation of fractional derivatives is used: the convolution kernel is replaced by a finite number of memory variables that satisfy local-in-time ordinary differential equations. The coefficients of the diffusive representation follow from an optimization procedure of the dispersion relation. Then, various methods of scientific computing are applied: the propagative part of the equations is discretized using a fourth-order ADER scheme, whereas the diffusive part is solved exactly. An immersed interface method is implemented to discretize the geometry on a Cartesian grid, and also to enforce the jump conditions at interfaces. Numerical experiments are proposed in various realistic configurations.