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


Geometry
---------

The field studied is a :math:`3m` bar.


.. image:: images/1000000000000168000000A7B0D6772D8EDCB228.gif
    :width: 3.75in
    :height: 1.7398in


.. _RefImage_1000000000000168000000A7B0D6772D8EDCB228.gif:

Coordinates of points :math:`(m)`:

:math:`A(0;0)` :math:`C(3;0.1)`

:math:`B(3;0)` :math:`D(0;0.1)`


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

Here we take data that leads to an almost unitary problem. The units then no longer have any physical meaning.

Gravity is taken in the positive :math:`x` direction (which therefore corresponds to the vertical axis).


.. csv-table::

    "Liquid water", "Density :math:`({\mathrm{kg.m}}^{-3})`
    Viscosity :math:`(\mathrm{Pa.}s)` ", 1"
    1"
    "Gas", "Viscosity :math:`(\mathrm{Pa.}s)` ", "1"
    "Homogenized parameters", "Gravity :math:`(\mathrm{m.}{s}^{-2})`
    Permeability :math:`K({m}^{2})`
    Porosity
    Sorption isotherm
    Relative permeability", ":math:`g=(\mathrm{9,81 };\mathrm{0 };0)`
    1
    0.5
    :math:`{S}_{\text{we}}=\frac{1}{{\left[1+{(\frac{{P}_{c}}{1})}^{\mathrm{1,5}}\right]}^{1/3}}`
    :math:`\begin{array}{}{\text{kr}}_{w}(S)=1\\ {\text{kr}}_{\text{gz}}(S)=1\end{array}`"



Boundary and initial conditions
-----------------------------------

There is no flow everywhere. Initially, the medium is desaturated with a saturation of :math:`S=\mathrm{0,5}` over the entire domain, which corresponds to capillary pressure:

 :math:`\mathrm{Pc}=\mathrm{3,6}\mathrm{Pa}`.

An initial gas pressure of :math:`1\mathrm{Pa}` is taken.


No time
------------

We model :math:`1s` in the following way:


* from :math:`0` to :math:`\mathrm{0,1}s`: 5 steps of time


* from :math:`\mathrm{0,1}` to :math:`1s`: 9 steps of time


Benchmark solution
---------------------

In the steady state, hydrostatic equilibrium must be achieved.

So we have to :math:`{\mathrm{\Delta P}}_{\text{lq}}=\rho \text{.}g\text{.}\mathrm{\Delta x}`

With the data we have, it therefore gives us :math:`{\mathrm{\Delta P}}_{\text{lq}}=\mathrm{9,}\text{81}\ast 3=\mathrm{29,43}\mathrm{Pa}`