Reference problem
=====================


Geometry of the problem
---------------------

It is a block with height :math:`\mathit{LZ}=10m`, length :math:`\mathit{LX}=10m`, and width :math:`\mathit{LY}=2m`. This block has a cohesive interface-type discontinuity (a non-meshed interface that is introduced into the model through level-sets using the DEFI_FISS_XFEM operator). It is identified by the normal level-set of equation :math:`\mathit{lsn}=Z-5` and crosses the block entirely in the horizontal direction by dividing it into two identical sub-blocks. Points :math:`A(\mathrm{0,}\mathrm{0,}5)`, :math:`A\text{'}(\mathrm{0,}\mathrm{2,}5)`, :math:`B(\mathrm{3,}\mathrm{0,}5)`, and :math:`B\text{'}(\mathrm{3,}\mathrm{2,}5)` will be used for the imposition of boundary conditions and the evaluation of the quantities tested.

The geometry of the block is represented in the figure.

**Figure** 1.1-a **: Problem geometry**


.. image:: images/10000000000002DA000002359C6D1F50EFD52538.jpg
    :width: 6.4035in
    :height: 4.9555in


.. _RefImage_10000000000002DA000002359C6D1F50EFD52538.jpg:


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 '. The cohesive model type is' STANDARD 'and the cohesive law used is' CZM_OUV_MIX '.


.. 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:`0.2`
    :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})` "," :math:`1`
    :math:`\mathrm{2,5}`
    :math:`{10}^{\text{-18}}`"
    "Parameters of the cohesive law", "Critical constraint :math:`{\mathrm{\sigma }}_{c}(\mathit{en}\mathit{MPa})`
    Cohesive energy :math:`{G}_{c}(\mathit{en}\mathit{Pa}\mathrm{.}m)`
    Increase coefficient :math:`r` "," :math:`0.5`
    :math:`9000`
    :math:`100`"


**Table** 1.2-1 **: 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:`\mathrm{\varphi }\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}\mathrm{0,1}`.


Boundary conditions and loads
-------------------------------------

The Dirichlet conditions that are applied are:


* the following moves :math:`x` are stuck in on the left edge of the domain,


* the movements following :math:`y` and :math:`z` are blocked on the lower and upper edges of the domain.

Also, a punctual flow of fluid :math:`Q=0.025\mathit{kg}\mathrm{.}{m}^{-1}\mathrm{.}{s}^{-1}` is injected into the cohesive interface on the left edge of the domain for a period of time :math:`t=10s`.