Numerical modeling of 1D transient poroelastic waves in the low-frequency range

Propagation of transient mechanical waves in porous media is numerically investigated in 1D. The framework is the linear Biot model with frequency-independent coefficients. The coexistence of a propagating fast wave and a diffusive slow wave makes numerical modeling tricky. A method combining three numerical tools is proposed: a fourth-order ADER scheme with time-splitting to deal with the time-marching, a space-time mesh refinement to account for the small-scale evolution of the slow wave, and an interface method to enforce the jump conditions at interfaces. Comparisons with analytical solutions confirm the validity of this approach.

[1]  Claus-Dieter Munz,et al.  Lax–Wendroff-type schemes of arbitrary order in several space dimensions , 2006 .

[2]  H. Kreiss,et al.  Stability Theory of Difference Approximations for Mixed Initial Boundary Value Problems. II , 1972 .

[3]  Chenggang Zhao,et al.  An explicit finite element method for Biot dynamic formulation in fluid-saturated porous media and its application to a rigid foundation , 2005 .

[4]  José M. Carcione,et al.  SOME ASPECTS OF THE PHYSICS AND NUMERICAL MODELING OF BIOT COMPRESSIONAL WAVES , 1995 .

[5]  L. Trefethen.  Instability of difference models for hyperbolic initial boundary value problems , 1984 .

[6]  Joël Piraux,et al.  A new interface method for hyperbolic problems with discontinuous coefficients: one-dimensional acoustic example , 2001 .

[7]  R. LeVeque,et al.  Adaptive Mesh Refinement Using Wave-Propagation Algorithms for Hyperbolic Systems , 1998 .

[8]  Jian-Fei Lu,et al.  Wave field simulation for heterogeneous porous media with singular memory drag force , 2005 .

[9]  Denis Matignon,et al.  Efficient solution of a wave equation with fractional-order dissipative terms , 2010, J. Comput. Appl. Math..

[10]  Boris Gurevich,et al.  Wave Propagation in heterogeneous, porous media: A velocity-stress, finite difference method; discussion and reply , 1996 .

[11]  M. Biot.  Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid. II. Higher Frequency Range , 1956 .

[12]  José M. Carcione,et al.  Note: Numerical Solution of the Poroviscoelastic Wave Equation on a Staggered Mesh , 1999 .

[13]  Qing Huo Liu,et al.  The application of the perfectly matched layer in numerical modeling of wave propagation in poroelastic media , 2001 .

[14]  Marsha Berger,et al.  Stability of interfaces with mesh refinement , 1985 .

[15]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[16]  Michael Dumbser,et al.  Fast high order ADER schemes for linear hyperbolic equations , 2004 .

[17]  Boris Gurevich,et al.  Interface conditions for Biot’s equations of poroelasticity , 1999 .

[18]  Patrick Joly,et al.  An Error Analysis of Conservative Space-Time Mesh Refinement Methods for the One-Dimensional Wave Equation , 2005, SIAM J. Numer. Anal..

[19]  Martin Schanz,et al.  Application of 3D time domain boundary element formulation to wave propagation in poroelastic solids , 2001 .

[20]  B Lenoach.  1D waves in a random poroelastic medium with large fluctuations , 1999 .

[21]  Abdelaâziz Ezziani,et al.  Modélisation mathématique et numérique de la propagation d'ondes dans les milieux viscoélastiques et poroélastiques , 2005 .

[22]  B. Gustafsson.  The convergence rate for difference approximations to mixed initial boundary value problems , 1975 .

[23]  M. Biot.  Theory of Propagation of Elastic Waves in a Fluid‐Saturated Porous Solid. I. Low‐Frequency Range , 1956 .

[24]  S. Kelly,et al.  Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid , 1956 .

[25]  Olivier Coussy,et al.  Acoustics of Porous Media , 1988 .

[26]  D. L. Johnson,et al.  The equivalence of quasistatic flow in fluid‐saturated porous media and Biot’s slow wave in the limit of zero frequency , 1981 .

[27]  Nanxun Dai,et al.  Wave propagation in heterogeneous, porous media: A velocity‐stress, finite‐difference method , 1995 .