1. Reference problem#
1.1. Geometry of the problem#
It is a column with height \(\mathit{LZ}=5m\), length \(\mathit{LX}=1m\), and width \(\mathit{LY}=1m\). In \(Z=\frac{\mathit{LZ}}{2}\), this column has an interface-type discontinuity. The column is thus entirely crossed by the discontinuity.
The geometry of the column is shown in the figure.
Figure 1.1-a : Problem geometry
1.2. Material properties#
The parameters given in the Table correspond to the parameters used for modeling in the hydromechanical coupled case. The coupling law used is” LIQU_SATU “. The parameters specific to this coupling law are given but have no influence on the solution (because we chose to take a uniformly zero pore pressure throughout the domain). Only elastic parameters have an influence on the solution of the pseudo-coupled problem.
Liquid (water) |
Viscosity \({\mu }_{w}(\mathit{en}\mathit{Pa.s})\) Compressibility module \(\frac{1}{{K}_{w}}(\mathit{en}{\mathit{Pa}}^{\text{-1}})\) Liquid density \({\rho }_{w}(\mathit{en}\mathit{kg}\mathrm{/}{m}^{3})\) |
\({5.10}^{\text{-10}}\) \(1\) |
Elastic parameters |
Young’s modulus \(E(\mathit{en}\mathit{MPa})\) Poisson’s ratio \(\nu\) Thermal expansion coefficient \(\alpha (\mathit{en}{K}^{\text{-1}})\) |
\(0\) \(0\) |
Coupling parameters |
Biot coefficient \(b\) Initial homogenized density \({r}_{0}(\mathit{en}\mathit{kg}\mathrm{/}{m}^{3})\) Intrinsic permeability \({K}^{\text{int}}(\mathit{en}{m}^{2})\) |
\(\mathrm{2,5}\) \({\mathrm{1,01937}}^{\text{-9}}\) |
Parameters of the cohesive law |
Critical constraint \({\mathrm{\sigma }}_{c}(\mathit{en}\mathit{MPa})\) Cohesive energy \({G}_{c}(\mathit{en}\mathit{Pa}\mathrm{.}m)\) Increase coefficient \(r\) |
\(900\) \(10\) |
Table 1.2-1 : Material Properties
On the other hand, the forces associated with gravity (in the equation for the conservation of momentum) are neglected. The reference pore pressure is taken to be zero \({p}_{1}^{\text{ref}}=0\mathit{MPa}\) and the porosity of the material is \(\varphi =\mathrm{0,15}\).
1.3. Boundary conditions#
The following Dirichlet conditions apply:
within the set of the domain, movements are blocked at zero (\({u}_{\text{x}}=0\), \({u}_{\text{y}}=0\) and \({u}_{\text{z}}=0\) in the three-dimensional case),
on the [ABCD] and [EFGH] faces, the pore pressure is blocked at zero,
in the cohesive interface, fluid pressure \({p}_{\text{f}}=10\mathit{MPa}\) is imposed.
The initial pore pressure in the matrix is zero. The test is carried out over a total period of time \(t=10s\). Parameter \(\mathrm{\theta }\) is taken to be equal to \(\mathrm{0,56999999}\).