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


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

It is a square block with side :math:`L=10m`. This block has two cohesive interface discontinuities (non-meshed interface introduced into the model in the form of a level curve (level-set) using the operator DEFI_FISS_XFEM). The center of the square and the origin of the coordinate system :math:`\mathit{Oxy}` are confused. The first is identified by the normal level-set of equation :math:`{\mathit{lsn}}_{1}=Y-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+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:`\{\begin{array}{c}X=-0.4\\ Y=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(\mathrm{5,}2.7)`, :math:`B(\mathrm{5,}-2.7)`, :math:`C(-\mathrm{1,}-\mathrm{0,3})`, and :math:`U(\mathrm{5,}0)` 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/10000000000002AC0000022BF92D7437A9B37830.jpg
    :width: 5.6374in
    :height: 4.5736in


.. _RefImage_10000000000002AC0000022BF92D7437A9B37830.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' MORTAR 'and the cohesive law used is' CZM_LIN_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:`0.25`
    :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:`\mathrm{0,8}`
    :math:`\mathrm{2,5}`
    :math:`{10}^{\text{-15}}`"
    "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:`0.11`
    :math:`50`
    :math:`2`"


**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 :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:


* blocking movements following :math:`x` on the right edge of the domain;


* blocking movements following :math:`y` on the lower and upper edges of the domain;


* blocking vertical movements in the middle block;


* blocking of movements following :math:`y` to point :math:`U`.

A punctual flow of fluid :math:`Q=0.04\mathit{kg}\mathrm{.}{s}^{-1}` is injected into the cohesive interfaces at points :math:`A` and :math:`B` for a period of time :math:`t=50s`.