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


Presentation
------------

In this test case, we study the hydraulic behavior of a saturated porous medium constituted by a single fluid: water in its liquid phase. The associated law of fluid behavior is of type LIQU_SATU. The modeling is hydromechanical (HM). The fluid is incompressible and the medium is infinitely rigid. It is therefore a stationary problem involving an analytical solution. The objective of this test is to validate the correct consideration of a hydraulic flow boundary condition in the case of axisymmetric modeling.


Geometry
---------

A bar of length :math:`L=\mathrm{1m}` is represented. Its width l is not involved in the analytical solution because the problem is purely 1D. Here we take :math:`l=\mathrm{0,2}m`. Point N1 corresponds to the coordinate point :math:`(\mathrm{0,0})\mathrm{.}`


.. image:: images/10000200000000C400000135C3959BE754F1F575.png
    :width: 1.3791in
    :height: 2.3673in


.. _RefImage_10000200000000C400000135C3959BE754F1F575.png:

Illustration 1: vertical bar in vertical positioning


Material properties
----------------------


.. csv-table::

    "Liquid water", "Density :math:`{\rho }_{l}({\mathit{kg.m}}^{-3})`
    Dynamic viscosity of liquid water :math:`{\mu }_{l}(\mathit{Pa.s})`
    Compressibility :math:`{K}_{w}({\mathit{Pa}}^{-1})` "," :math:`{10}^{3}`
    :math:`0.001`
    :math:`0`"
    "Solid", "Drained Young's Module :math:`E(\mathrm{Pa})`
    Poisson's Ratio :math:`\nu` "," :math:`7.5{10}^{15}`
    :math:`0.3`"
    "Reference state", "Porosity :math:`\phi`
    Temperature :math:`{T}^{\text{ref}}(K)`
    Liquid pressure :math:`{P}^{\text{ref}}(\mathit{Pa})`
    Vapor pressure :math:`{\mathit{Pv}}^{\text{ref}}(\mathit{Pa})` "," :math:`0.2`
    :math:`293`
    :math:`0`
    :math:`0.1`"
    "Constants", "Ideal gas constant R", ":math:`8.32`"
    "Homogenized coefficients", "Homogenized density :math:`{r}_{0}({\mathit{kg.m}}^{-3})`
    Biot coefficient b
    Intrinsic permeability :math:`{K}_{\text{int}}({m}^{2})` "," :math:`2200`
    :math:`0.6`
    :math:`{K}_{\text{int}}={10}^{-12}`"


**Table** 1.3-1 **: Material data**

The gravity of water is overlooked.


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

On all sides

     blocked trips :math:`{u}_{x}={u}_{y}={u}_{z}=0`

Upper edge:

     water flow: :math:`q=1{\mathit{kg.s}}^{-1}\mathrm{.}{m}^{-2}`

Lower edge:

     Liquid pressure: :math:`{P}_{\mathit{lq}}={P}_{0}=1\mathit{atm}`

Side edges:

  zero flow


Initial conditions
--------------------

The initial pressure is :math:`1\mathit{atm}`. All other fields are null.