Dynamic Behavior of Saturated Poroelastic Continuum by Simplified Formulation of Biot’s Theory

  • Dan HuEmail author
  • Kaiyin Zhang
  • Fen Li
Original Paper


The behavior of saturated, porous media under dynamic or quasi-static loads was firstly formulated by Biot. Based on Biot’s theory, various possible simplifications have been proposed such as neglecting the second time derivative of the relative displacement. In addition, an even more simplified version has been proposed where the inertial terms in the generalized Darcy’s law are neglected as well. This paper aims to explore the advantages and limitations of that simplified formulation with the help of a linear one-dimensional (1D) poroelastic column subjected to periodic dynamic loading. Two non-dimensional parameters which are related to the permeability, excitation frequency, and material properties are introduced, and the analytical solutions are obtained and depicted graphically in cases of varied permeability and frequencies. Subsequently, a schematic diagram is developed to serve as guideline to determine whether the simplification is tenable, or the only meaningful solution is available by complete Biot’s theory.


Biot’s theory Simplified formulation FLAC3D Non-dimensional parameters 

List of Symbols


Total stress tensor (tension positive)

\( \alpha \)

Biot coefficient


Kronecker delta


Strain tensor

\( M \)

Biot modulus



\( \rho \)

Bulk density of mixture


Gravity acceleration

\( w_{i} \)

Relative displacement

\( q_{i} \)

Specific flux


Apparent mass density

\( \varOmega \)

\( M/\left( {E + \alpha^{2} M} \right) \)

\( \beta_{2} \)

\( \rho_{a} /\rho \)

\( ND_{1} \)

\( 2\rho \kappa T/\left( {n\pi \hat{T}^{2} } \right) \)


\( 2\pi /\omega \)

\( V_{c}^{2} \)

\( \left( {E + \alpha^{2} M} \right)/\rho \)

\( \sigma_{ij}^{\prime } \)

Effective stress tensor (tension positive)

\( p_{f} \)

Pore pressure (compression positive)

\( \lambda \), \( \mu \)

Lame constants

\( u_{i} \)

Solid displacement tensor

\( \zeta \)

Variation of fluid content

\( e \), \( \theta \)

Volumetric strains of solid and fluid

\( \rho_{\text{s}} \), \( \rho_{\text{f}} \)

Real density of solid and fluid

\( \kappa \)

Intrinsic permeability

\( k \)

Permeability coefficient

\( \mu_{\text{f}} \)

Fluid viscosity

\( \omega \)

Angular frequency

\( \beta_{1} \)

\( \rho_{f} /\rho \)

\( \bar{z} \)


\( ND_{2} \)

\( \pi^{2} \left( {\hat{T}/T} \right)^{2} \)

\( \hat{T} \)

\( 2L/V_{\text{c}} \)

E, \( \nu \)

Elastic modulus and Poisson ratio


Supplementary material

40098_2019_354_MOESM1_ESM.m (21 kb)
Supplementary material 1 (M 21 kb)
40098_2019_354_MOESM2_ESM.m (7 kb)
Supplementary material 2 (M 6 kb)


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Copyright information

© Indian Geotechnical Society 2019

Authors and Affiliations

  1. 1.School of TransportationWuhan University of TechnologyWuhanChina

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