Finite difference modeling of Biot's poroelastic equations at seismic frequencies

Abstract

[1] Across the seismic band of frequencies (loosely defined as <10 kHz), a seismic wave propagating through a porous material will create flow in the pore space that is laminar; that is, in this low-frequency ''seismic limit,'' the development of viscous boundary layers in the pores need not be modeled. An explicit time stepping staggered-grid finite difference scheme is presented for solving Biot's equations of poroelasticity in this low-frequency limit. A key part of this work is the establishment of rigorous stability conditions. It is demonstrated that over a wide range of porous material properties typical of sedimentary rock and despite the presence of fluid pressure diffusion (Biot slow waves), the usual Courant condition governs the stability as if the problem involved purely elastic waves. The accuracy of the method is demonstrated by comparing to exact analytical solutions for both fast compressional waves and slow waves. Additional numerical modeling examples are also presented.

Bridging aero-fracture evolution with the characteristics of the acoustic emissions in a porous medium

“…where the part before the pressure gradient of Equation (16) can be described as diffusivity constant of overpressure in a porous medium, corresponding to the slow Biot wave [79]:…”

Acoustic Attenuation in Self-Affine Porous Structures

“…numerically by performing finite-difference simulations of the above experiment. Details of how the finite-difference algorithm works are given by Masson and Pride [13,14]. The numerical simulations are based on the laws of poroelasticity [15,16] that provide a continuum description allowing for fluid-pressure changes and fluid flow in addition to the elastic deformation and acceleration of the material.…”

On the propagation of a disturbance in a heterogeneous, deformable, porous medium saturated with two fluid phases

“…Also, numerical methods have difficulty modeling the wide range of behaviors in the coupled multiphase problem, which can include hyperbolic elastic wave propagation as well as fluid diffusion, involving a broad range of time scales: from milliseconds to hours or even days. Numerical methods tailored to seismic frequencies can improve the computational efficiency (Masson et al, 2006) but still face challenges in treating multiple fluid phases and 3D problems. Finally, numerical methods do not provide explicit expressions for observed quantities such as the arrival time of a propagating disturbance or its amplitude.…”