Finite-difference analysis of fully dynamic problems for saturated porous media

Finite-difference methods, using staggered grids in space, are considered for the numerical approximation of fully dynamic poroelasticity problems. First, a family of second-order schemes in time is analyzed. A priori estimates for displacements in discrete energy norms are obtained and the corresponding convergence results are proved. Numerical examples are given to illustrate the convergence properties of these methods. As in the case of an incompressible fluid and small permeability, these schemes suffer from spurious oscillations in time, a first order scheme is proposed and analyzed. For this new scheme a priori estimates and convergence results are also given. Finally, numerical examples in one and two dimensions are presented to show the good monotonicity properties of this method.

[1]  Petr N. Vabishchevich,et al.  Finite difference analysis of a double‐porosity consolidation model , 2012 .

[2]  A. Samarskii.  The Theory of Difference Schemes , 2001 .

[3]  Hester Bijl,et al.  A Unified Method for Computing Incompressible and Compressible Flows in Boundary-Fitted Coordinates , 1998 .

[4]  Tongchun Li,et al.  Stabilized finite elements for harmonic soil dynamics problems near the undrained-incompressible limit , 1997 .

[5]  F. Gaspar,et al.  Staggered grid discretizations for the quasi-static Biot's consolidation problem , 2006 .

[6]  F. Gaspar,et al.  A finite difference analysis of Biot's consolidation model , 2003 .

[7]  F. Harlow,et al.  Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface , 1965 .

[8]  O. C. Zienkiewicz,et al.  On the stable finite element procedures for dynamic problems of saturated porous media , 2004 .

[9]  K. Y. Lam,et al.  A numerical study of wave propagation in a poroelastic medium by use of localized differential quadrature method , 2004 .

[10]  Maosong Huang,et al.  A New Algorithm for the Coupled Soil–Pore Fluid Problem , 1993 .

[11]  O. C. Zienkiewicz,et al.  DRAINED, UNDRAINED, CONSOLIDATING AND DYNAMIC BEHAVIOUR ASSUMPTIONS IN SOILS , 1980 .

[12]  Petr N. Vabishchevich,et al.  A stabilized method for a secondary consolidation Biot's model , 2008 .

[13]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[14]  Patrick Saint Macary.  Analyse mathématique de modèles de diffusion en milieu poreux élastique , 2004 .

[15]  Hélène Barucq,et al.  On nonlinear Biot's consolidation models , 2005 .

[16]  M. Biot.  General solutions of the equations of elasticity and consolidation for a porous material , 1956 .

[17]  M. Biot.  General Theory of Three‐Dimensional Consolidation , 1941 .

[18]  Dongho Shin,et al.  Inf-sup conditions for finite-difference approximations of the stokes equations , 1997, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.