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

Abstract

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.

Appendix: Analytical Solution in Time Domain by Convolution Quadrature Method

Generally, the response in time domain can be calculated using the convolution integral as below,

$$ u_{z} \left( {t,z} \right) = \mathop \int \limits_{0}^{t} {\mathcal{L}}^{ - 1} \left\{ {\tilde{u}_{z} \left( {s,z} \right)} \right\}\left( {\tau ,z} \right)f\left( {t - \tau } \right){\text{d}}\tau $$

(24)

$$ p_{f} \left( {{\text{t}},{\text{z}}} \right) = \mathop \int \limits_{0}^{t} {\mathcal{L}}^{ - 1} \left\{ {\tilde{p}_{f} \left( {s,z} \right)} \right\}\left( {\tau ,z} \right)f\left( {t - \tau } \right){\text{d}}\tau $$

(25)

where ${\mathcal{L}}^{ - 1}$ is the inverse Laplace transform operator and $s$ is the complex Laplace variable.

We found the response function in the convolution integral (24)–(25) is only available in Laplace domain and force function in time domain and a numerical inverse Laplace transformation is necessary. It is preferable to take the “Convolution Quadrature Method” proposed by Lubich, and this numerical method approximates the convolution integral numerically by a quadrature formula,

$$ u_{z} \left( {n\Delta t} \right) = \mathop \sum \limits_{k = 0}^{n} \omega_{n - k} \left( {\Delta t} \right)f\left( {k\Delta t} \right),\quad n = 0,1, \ldots ,N $$

(26)

where $N$ is divide time t by time increment $\Delta t$ and weights $\omega_{n - k} \left( {\Delta t} \right)$ are determined with the help of Laplace-transformed impulse response function $\hat{u}_{z} \left( {s,z} \right)$ and a linear multistep method $\gamma \left( s \right)$ published by Lubich [15, 16], as

$$ \omega_{n} \left( {\Delta t} \right) = \frac{{{\mathcal{R}}^{ - n} }}{L}\mathop \sum \limits_{m = 0}^{L - 1} \hat{u}_{z} \left( {\frac{{\gamma \left( {{\mathcal{R}}e^{{im\frac{2\pi }{L}}} } \right)}}{\Delta t}} \right){\text{e}}^{{ - inm\frac{2\pi }{L}}} $$

(27)

where ${\mathcal{R}}$ is the radius of a circle in the domain of analyticity of $\hat{u}_{z}$ and function $\gamma \left( s \right)$ is the polynomial of the multistep method.

The used parameters can be found in Ref. [13], i.e., $\gamma \left( s \right) = {\raise0.7ex\hbox{$3$} \!\mathord{\left/ {\vphantom {3 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}} - 2s + {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}s^{2}$, $L = N$, ${\mathcal{R}}^{N} = \sqrt \varepsilon$, $\varepsilon = 10^{ - 10}$, which is tested against the time domain solution for the particular case with infinite permeability and has produced favorable comparison with small time increment $\Delta t$.