1. Reference problem#
1.1. Geometry#
The mesh of the poroelastic column is given opposite. The propagation axis is directed along \((\mathit{OY})\). The total height of the column is \(25m\), for \(0.4m\) in width. The first 5 meters at the bottom are discretized by \(1m\) thick elements while the following \(20m\) are discretized by \(0.2m\) thick elements. The entire column thus contains 105 items.
Image 1.1.1: Column mesh
1.2. Material properties#
The table below shows the material properties of coupled hydromechanical modeling. The porous elastic material is assumed to be linear and fully saturated with water.
Based on the following material properties, the speed of the compression waves is given by the relationship:
\({C}_{p}=\sqrt{\frac{\lambda +2\mu }{{\rho }_{\mathit{homo}}}}=\mathrm{534,14}{\mathit{m.s}}^{-1}\)
where \(\lambda\) and \(\mu\) are Lamé’s modules.
Liquid water |
Density \({\rho }_{l}\) Compressibility \({K}_{l}\) Viscosity \(\nu\) |
\(\mathrm{0,1}\mathit{MPa}\) \(\mathrm{0,001}\) |
Homogenized parameters |
Intrinsic permeability \({K}^{\text{int}}\) Porosity \(\varphi\) Homogenized density \({\rho }_{\mathit{homo}}\) |
\(\mathrm{0,23}\) \(2105{\mathit{kg.m}}^{-3}\) |
Linear elastic solid skeleton |
Compressibility module \(K\) Shear module \(G\) |
\(\mathrm{215,6}\mathit{MPa}\) |
Table 1.2-1: Hydromechanical properties of the poroelastic column
1.3. Boundary and initial conditions#
It is desired to propagate a compression wave from bottom to top in the column with a blocking of its upper part so that there is reflection of the wave at this location and reverse propagation from top to bottom. An absorbent border (paraxial element) is placed at the base of the column in order to completely dampen the reflected wave.
Since the column is subject to gravity, geostatic conditions in terms of effective stresses and in hydraulic pressure initially pre-exist in the column. The initial balancing of the column is not obvious and requires the following procedure:
the base of the column, flanked by the paraxial element (in red), is blocked using very stiff springs. A prior static calculation (**figure 1*) makes it possible to recover a field of stresses and hydraulic pressure in balance with the boundary and loading conditions;
we calculate the nodal forces (resulting from both mechanical and hydraulic forces, in green) in the springs (**figure 2*);
These nodal forces are reinjected at the base where the springs are simultaneously deactivated by giving them very low stiffness (**figure 3*). The boundary conditions thus defined make it possible to produce a perfectly balanced initial hydromechanical state;
The dynamic calculation can take place by injecting the wave from the paraxial element (**figure 4*);
In order to account for the invariance by horizontal translation (1D problem), lateral displacements are joined together by a LIAISON_DDL type connection.
Figure 1.3.1: Initial column balancing procedure
The boundary, initial, and loading conditions are summarized in the following tables:
Boundary conditions |
||
\(\mathit{HAUT}\) |
MECA_IMPO |
\(\mathit{DX}=\mathit{DY}=\mathit{DZ}=0\) |
\(\mathit{COTE}\) |
LIAISON_DDL |
\({\mathit{DX}}_{\mathit{gauche}}={\mathit{DX}}_{\mathit{droit}}\) \({\mathit{DY}}_{\mathit{gauche}}={\mathit{DY}}_{\mathit{droit}}\) \({\mathit{DZ}}_{\mathit{face}}={\mathit{DZ}}_{\mathit{arrière}}\) |
\(\mathit{BAS}\) |
FORC_NODA |
|
Table 1.3-1: boundary conditions
Initial conditions |
||
\(\mathit{COLONNE}\) |
Geostatic |
\(\mathit{SIXX}\mathrm{=}\mathit{SIYY}\mathrm{=}\mathit{SIZZ}\mathrm{=}(1\mathrm{-}\phi )({\rho }_{g}\mathrm{-}{\rho }_{l})\mathit{gY}\) \(\mathit{SIP}\mathrm{=}\mathrm{-}\mathit{PRE1}\mathrm{=}{\rho }_{l}\mathit{gY}\) |
Table 1.3-2: initial conditions
Loading |
||
\(\mathit{COLONNE}\) |
AFFE_CHAR_MECA: PESANTEUR |
\(g=-\mathrm{9,81}{\mathit{m.s}}^{-2}\) according to \((\mathit{OY})\) |
\(\mathit{BAS}\) |
AFFE_CHAR_MECA_F: ONDE_PLANE |
|
Table 1.3-3: loads
1.4. Plane wave#
The plane compression wave is a Ricker type wavelet whose profile is given below. It is important to note that the signal should be introduced in the form of a speed as a function of time.
Figure 1.4.1: Incident wavelet (Ricker)