Numerical modelling of saturated boundless media with infinite elements

Abstract

Based on the elastic theory assumptions and averaging theories, an infinite element boundary which is frequency independent is derived for saturated soil media. The infinite element development is based on mapping functions and viscous layers for damping propagating waves of both solid and water phases. The newly developed infinite element is able to simulate the boundaries of fully saturated soil medium considering both soil displacement and fluid pressures. The main point is that the fluid pressure gradient at infinity is taken to equal to zero thus enabling the realistic consideration of the fluid pressures in numerical simulations. In numerical modelling, the general finite element software ANSYS using its User Programmable Features is used. Related comparisons are done with references. In simulation of propagating waves, the numerical approach is verified considering different models. The verification models include wave propagation through a saturated soil column. The implementation of the newly developed infinite element is done by simulation of an earth fill dam boundaries while the dam body is simulated as a three-phase medium. The obtained results from simulations show promising results and are thoroughly discussed.

Appendix

The mass matrices are given as follows:

$$ {\mathbf{M}} = \smallint {\mathbf{N}}_{{\mathbf{u}}} \left[ {\rho_{s} \left( {1 - n} \right) + nS_{w} \rho_{w} } \right]{\mathbf{N}}_{{\mathbf{u}}} \;{\text{d}}\Omega $$

$$ {\mathbf{M}}_{{\varvec{f}}} = \smallint \nabla {\mathbf{N}}_{{\mathbf{p}}}^{{\text{T}}} \frac{{{\mathbf{k}}k_{rf} }}{{\eta_{f} }}{\mathbf{N}}_{{\mathbf{u}}} \;{\text{d}}\Omega $$

The coupling matrix follows as:

$$ {\mathbf{C}}_{{{\text{sf}}}} = \smallint {\mathbf{N}}_{p}^{T} \alpha S_{f} m^{T} {\mathbf{LN}}_{{\mathbf{u}}} \;{\text{d}}\Omega $$

The compressibility matrix is given as:

$$ {\mathbf{P}}_{{{\text{ff}}}} = \smallint {\mathbf{N}}_{p}^{T} \left[ {\frac{{S_{f} }}{{K_{f} }} + \left( {\alpha - n} \right)\frac{{S_{f} }}{{K_{s} }}\left( {S_{f} + p_{c} \frac{{\partial S_{f} }}{{\partial p_{c} }}} \right) - n\frac{{\partial S_{f} }}{{\partial p_{c} }}} \right]{\mathbf{N}}_{{\mathbf{p}}} \;{\text{d}}\Omega $$

The permeability matrix can be written as:

$$ {\mathbf{H}}_{{{\varvec{ff}}}} = \smallint \nabla {\mathbf{N}}_{{\mathbf{p}}}^{{\text{T}}} \frac{{{\mathbf{k}}k_{rf} }}{{\eta_{f} }}\nabla {\mathbf{N}}_{{\mathbf{p}}} \;{\text{d}}\Omega $$

The domain forces follow as:

$$ f_{u} = \smallint {\mathbf{N}}_{{\text{u}}} \left[ {\rho_{s} \left( {1 - n} \right) + nS_{f} \rho_{f} } \right]{\mathbf{gN}}_{{\mathbf{u}}} \;{\text{d}}\Omega $$

$$ {\varvec{f}}_{f} = \smallint {\mathbf{N}}_{{\mathbf{p}}}^{{\text{T}}} \frac{{{\mathbf{k}}k_{rf} }}{{\eta_{f} }}\rho_{f} {\mathbf{g}}\;{\text{d}}\Omega $$