Reference problem ===================== Geometry of the 2D problem (modeling A) ----------------------------------------- It is a square block with side :math:`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 :math:`{\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 :math:`{\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 :math:`{\mathit{lsn}}_{1}<0`. The junction point between the two interfaces verifies :math:`{\mathit{lsn}}_{1}={\mathit{lsn}}_{2}=0` and has coordinates :math:`(-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 :math:`A(\mathrm{5,}0)`, :math:`B(-\mathrm{3,}-1.3)`, :math:`C(\mathrm{3,}-1.7)` and :math:`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. .. image:: images/10000000000003A2000002BEEE3439C78515741D.jpg :width: 3.8547in :height: 3.1528in .. _RefImage_10000000000003A2000002BEEE3439C78515741D.jpg: **Figure** 1.1-a **: 2D Problem Geometry** Geometry of the 3D problem (B modeling) ----------------------------------------- It is a block with height :math:`\mathit{LZ}=10m`, length :math:`\mathit{LX}=10m`, and width :math:`\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 :math:`{\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 :math:`{\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 :math:`{\mathit{lsn}}_{1}<0`. The junction curve between the two interfaces verifies :math:`{\mathit{lsn}}_{1}={\mathit{lsn}}_{2}=0` and has the equation :math:`\{\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 :math:`{A}_{1}(\mathrm{5,}-\mathrm{1,}0)`, :math:`{A}_{2}(\mathrm{5,}\mathrm{1,}0)` :math:`B(-\mathrm{3,}-\mathrm{1,}-\mathrm{1,3})`, :math:`C(\mathrm{3,}-\mathrm{1,}-1.7)` and :math:`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. .. image:: images/10000000000003B30000030FD7278FECF026EF1C.jpg :width: 5.1681in :height: 4.3866in .. _RefImage_10000000000003B30000030FD7278FECF026EF1C.jpg: **Figure** 1.2-a **: Problem geometry** **3D** 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 '. .. csv-table:: "", "", "" "Liquid (water)", "Viscosity :math:`{\mu }_{w}(\mathit{en}\mathit{Pa.s})` Compressibility module :math:`\frac{1}{{K}_{w}}(\mathit{en}{\mathit{Pa}}^{\text{-1}})` Liquid density :math:`{\rho }_{w}(\mathit{en}\mathit{kg}\mathrm{/}{m}^{3})` "," :math:`{10}^{\text{-3}}` :math:`{5.10}^{\text{-10}}` :math:`1`" "Elastic parameters", "Young's modulus :math:`E(\mathit{en}\mathit{MPa})` Poisson's Ratio :math:`\nu` Thermal expansion coefficient :math:`\alpha (\mathit{en}{K}^{\text{-1}})` "," :math:`5800` :math:`0` :math:`0`" "Coupling parameters", "Biot coefficient :math:`b` Initial homogenized density :math:`{r}_{0}(\mathit{en}\mathit{kg}\mathrm{/}{m}^{3})` Intrinsic permeability :math:`{K}^{\text{int}}(\mathit{en}{m}^{2}/s)` "," :math:`1` :math:`\mathrm{2,5}` :math:`{\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 :math:`{p}_{1}^{\text{ref}}=0\mathit{MPa}` and the porosity of the material is :math:`\varphi =\mathrm{0,15}`. We take :math:`\nu =0` in order to have a one-dimensional problem. 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 :math:`x` are blocked throughout the domain. The problem is thus unidirectional following :math:`y`, * the movements following :math:`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 :math:`A` is embedded. Neuman's conditions are: * a constant distributed mechanical pressure :math:`P=10\mathit{MPa}` is applied to each of the lips of the interfaces, * the pore pressure is set to :math:`{p}_{1}=0.2\mathit{MPa}` in the lower block, * the pore pressure is set to :math:`{p}_{2}=0.4\mathit{MPa}` in the intermediate block located between the two interfaces, * the pore pressure is set to :math:`{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 :math:`x` and the moves following :math:`y` are blocked throughout the domain. The problem is thus unidirectional following :math:`z`, * the movements following :math:`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 :math:`{A}_{1}` and :math:`{A}_{2}` are embedded. Neuman's conditions are: * a constant distributed mechanical pressure :math:`P=10\mathit{MPa}` is applied to each of the lips of the interfaces, * the pore pressure is set to :math:`{p}_{1}=0.2\mathit{MPa}` in the lower block, * the pore pressure is set to :math:`{p}_{2}=0.4\mathit{MPa}` in the intermediate block located between the two interfaces, * the pore pressure is set to :math:`{p}_{3}=0.6\mathit{MPa}` in the upper block. *