1. Reference problem#
1.1. Geometry of the 2D problem (modeling A)#
It is a square block with side \(L=10m\). This block has two interface-type discontinuities (non-meshed interface that is introduced into the model through level-sets using the operator DEFI_FISS_XFEM). The first is identified by the normal level-set of equation \({\mathit{lsn}}_{1}=Y-\mathrm{0.5X}-0.2\) and crosses the entire block in the horizontal direction. The second interface is identified by the normal level-set from equation \({\mathit{lsn}}_{2}=Y+\mathrm{0.5X}+0.2\). It connects to the lower lip of the first interface. So the second interface only exists in the part of the block such as \({\mathit{lsn}}_{1}<0\). The junction point between the two interfaces verifies \({\mathit{lsn}}_{1}={\mathit{lsn}}_{2}=0\) and has coordinates \((-0.4,0)\). The domain is thus divided into 3 blocks, a lower block, an upper block and an intermediate block located between the two interfaces. Points \(A(\mathrm{5,}0)\), \(B(-\mathrm{3,}-1.3)\), \(C(\mathrm{3,}-1.7)\) and \(D(\mathrm{3,}1.7)\) will be used for the application of boundary conditions and the evaluation of the quantities tested.
The geometry of the block is represented in the figure.
Figure 1.1-a : 2D Problem Geometry
1.2. Geometry of the 3D problem (B modeling)#
It is a block with height \(\mathit{LZ}=10m\), length \(\mathit{LX}=10m\), and width \(\mathit{LY}=2m\). This block has two interface-type discontinuities (non-meshed interface that is introduced into the model through level-sets using the operator DEFI_FISS_XFEM). The first is identified by the normal level-set of equation \({\mathit{lsn}}_{1}=Z-\mathrm{0.5X}-0.2\) and crosses the entire block in the horizontal direction. The second interface is identified by the normal level-set from equation \({\mathit{lsn}}_{2}=Z+\mathrm{0.5X}+0.2\). It connects to the lower lip of the first interface. So the second interface only exists in the part of the block such as \({\mathit{lsn}}_{1}<0\). The junction curve between the two interfaces verifies \({\mathit{lsn}}_{1}={\mathit{lsn}}_{2}=0\) and has the equation \(\{\begin{array}{c}X=-0.4\\ Z=0\end{array}\). The domain is thus divided into 3 blocks, a lower block, an upper block and an intermediate block located between the two interfaces. Points \({A}_{1}(\mathrm{5,}-\mathrm{1,}0)\), \({A}_{2}(\mathrm{5,}\mathrm{1,}0)\) \(B(-\mathrm{3,}-\mathrm{1,}-\mathrm{1,3})\), \(C(\mathrm{3,}-\mathrm{1,}-1.7)\) and \(D(\mathrm{3,}-\mathrm{1,}1.7)\) will be used for the application of boundary conditions and the evaluation of the quantities tested.
The geometry of the block is represented in the figure.
Figure 1.2-a : Problem geometry 3D
1.3. Material properties#
The parameters given in the Table correspond to the parameters used for modeling in the hydro-mechanical coupled case. The coupling law used is” LIQU_SATU “.
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 \({K}^{\text{int}}(\mathit{en}{m}^{2}/s)\) |
\(\mathrm{2,5}\) \({\mathrm{1,01937}}^{\text{-19}}\) |
Table 1.3-1 : Material Properties
On the other hand, the forces related to 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 \(\varphi =\mathrm{0,15}\).
We take \(\nu =0\) in order to have a one-dimensional problem.
1.4. Boundary conditions and loads#
2D case
The boundary conditions that can be applied to the domain are of two types:
Dirichlet-type conditions,
Neuman-type conditions.
Dirichlet’s conditions are:
the following moves \(x\) are blocked throughout the domain. The problem is thus unidirectional following \(y\),
the movements following \(y\) are blocked on the lower side and the upper side of the block,
in order to block the movements of the rigid body of the block located between the two interfaces, point \(A\) is embedded.
Neuman’s conditions are:
a constant distributed mechanical pressure \(P=10\mathit{MPa}\) is applied to each of the lips of the interfaces,
the pore pressure is set to \({p}_{1}=0.2\mathit{MPa}\) in the lower block,
the pore pressure is set to \({p}_{2}=0.4\mathit{MPa}\) in the intermediate block located between the two interfaces,
the pore pressure is set to \({p}_{3}=0.6\mathit{MPa}\) in the upper block.
3D case
The boundary conditions that can be applied to the domain are of two types:
Dirichlet-type conditions,
Neuman-type conditions.
Dirichlet’s conditions are:
the following moves \(x\) and the moves following \(y\) are blocked throughout the domain. The problem is thus unidirectional following \(z\),
the movements following \(z\) are blocked on the lower side and the upper side of the block,
in order to block the movements of the rigid body of the block located between the two interfaces, the points \({A}_{1}\) and \({A}_{2}\) are embedded.
Neuman’s conditions are:
a constant distributed mechanical pressure \(P=10\mathit{MPa}\) is applied to each of the lips of the interfaces,
the pore pressure is set to \({p}_{1}=0.2\mathit{MPa}\) in the lower block,
the pore pressure is set to \({p}_{2}=0.4\mathit{MPa}\) in the intermediate block located between the two interfaces,
the pore pressure is set to \({p}_{3}=0.6\mathit{MPa}\) in the upper block.