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


Geometry
---------

F

D

C

E

B

A

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


.. csv-table::

    "**Point**", ":math:`X` "," :math:`Y`"
    ":math:`A` ", "0.425", "-10"
    ":math:`B` ", "1,1225", "-10"
    ":math:`C` ", "1,1225", "0"
    ":math:`D` ", "0.425", "0"
    ":math:`E` ", "10", "-10"
    ":math:`F` ", "10", "0"


The part delimited by :math:`\mathit{ABCD}` will be called :math:`\mathit{BO}` and the part :math:`\mathit{BEFC}`, :math:`\mathit{BG}`.


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

The properties of the material are shown in the table below.


+----------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|Liquid water                                  |Density (:math:`{\mathit{kg.m}}^{\mathrm{-}3}`) Heat at constant pressure (:math:`{\mathit{J.K}}^{\mathrm{-}1}`) Coefficient of thermal expansion of liquid (:math:`{K}^{\mathrm{-}1}`) Dynamic viscosity of liquid water (:math:`\mathit{Pa.s}`)                                 |103 4180 10-4 10-3                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
+----------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|Gaz                                           |Specific heat (:math:`{\mathit{J.K}}^{\mathrm{-}1}`) Molar mass (:math:`{\mathit{kg.mol}}^{\mathrm{-}1}`)                                                                                                                                                                         |1000 0.02896 1.8. 10-5                                                                                                                                                                                                                                                                                                                                                                                                                                                |
+----------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|Sturdy (:math:`\mathit{BO}`)                  |Density (:math:`{\mathit{kg.m}}^{\mathrm{-}3}`) Drained Young's Modulus :math:`E` (:math:`\mathit{Pa}`) Poisson's Ratio                                                                                                                                                           |2670 1.9.1020 0.2                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
+----------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|Initial state (:math:`\mathit{BO}`)           |Porosity Temperature (:math:`K`) Gas Pressure (:math:`\mathit{Pa}`) Vapor Pressure (:math:`\mathit{Pa}`) Initial Capillary Pressure (:math:`\mathit{Pa}`)                                                                                                                         |0.35 293 1E5 2320 5.107 (:math:`S\mathrm{=}\mathrm{0,57}`)                                                                                                                                                                                                                                                                                                                                                                                                            |
+----------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
+----------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|Homogenized coefficients (:math:`\mathit{BO}`)|Homogenized density (:math:`{\mathit{kg.m}}^{\mathrm{-}3}`) Saturation Intrinsic permeability (:math:`{m}^{2}`) Relative liquid permeability () Relative liquid permeability Relative gas permeability (:math:`{\mathit{J.K}}^{\mathrm{-}1}`) Biot Thermal conductivities         |2670 :math:`S({P}_{C})\mathrm{=}0.99(1\mathrm{-}{6.10}^{\mathrm{-}9}{P}_{C})` 10-20                                                                                                                                                                                                                                                                                                                                                                                   |
+                                              +                                                                                                                                                                                                                                                                                  + .. image:: images/Object_2.svg                                                                                                                                                                                                                                                                                                                                                                                                                                       +
|                                              |                                                                                                                                                                                                                                                                                  |    :width: 65                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
+                                              +                                                                                                                                                                                                                                                                                  +    :height: 21                                                                                                                                                                                                                                                                                                                                                                                                                                                       +
|                                              |                                                                                                                                                                                                                                                                                  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
+                                              +                                                                                                                                                                                                                                                                                  +                                                                                                                                                                                                                                                                                                                                                                                                                                                                      +
|                                              |                                                                                                                                                                                                                                                                                  | .. image:: images/Object_4.svg                                                                                                                                                                                                                                                                                                                                                                                                                                       |
+                                              +                                                                                                                                                                                                                                                                                  +    :width: 65                                                                                                                                                                                                                                                                                                                                                                                                                                                        +
|                                              |                                                                                                                                                                                                                                                                                  |    :height: 21                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
+                                              +                                                                                                                                                                                                                                                                                  +                                                                                                                                                                                                                                                                                                                                                                                                                                                                      +
|                                              |                                                                                                                                                                                                                                                                                  | 482 1 :math:`\begin{array}{c}{\lambda }_{S}^{T}(S)\mathrm{=}\mathrm{0,35}\mathrm{\cdot }S\\ {\lambda }_{T}^{T}(S)\mathrm{=}\mathrm{0,6}\\ {\lambda }_{\mathit{CT}}^{T}(S)\mathrm{=}\mathrm{0,728}\end{array}`                                                                                                                                                                                                                                                        |
+----------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|Solid (BG)                                    |Density (:math:`{\mathit{kg.m}}^{\mathrm{-}3}`) Drained Young's Modulus :math:`E` (:math:`\mathit{Pa}`) Poisson's Ratio                                                                                                                                                           |2670 1.9.1020 0.2                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
+----------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|Initial state (BG)                            |Porosity Temperature (:math:`K`) Gas Pressure (:math:`\mathit{Pa}`) Vapor Pressure (:math:`\mathit{Pa}`) Initial Capillary Pressure (:math:`\mathit{Pa}`)                                                                                                                         |0.05 293 1E5 2320 7.107 (S=0.81)                                                                                                                                                                                                                                                                                                                                                                                                                                      |
+----------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|Homogenized coefficients (BG)                 |Homogenized density (:math:`{\mathit{kg.m}}^{\mathrm{-}3}`) Saturation Intrinsic permeability (:math:`{m}^{2}`) Relative liquid permeability () Relative liquid permeability () Saturation Intrinsic permeability () BioTthermal conductivity :math:`{\mathit{J.K}}^{\mathrm{-}1}`|2670 :math:`S({P}_{C})\mathrm{=}0.99(1\mathrm{-}{6.10}^{\mathrm{-}9}{P}_{C})` 10-19                                                                                                                                                                                                                                                                                                                                                                                   |
+                                              +                                                                                                                                                                                                                                                                                  + .. image:: images/Object_7.svg                                                                                                                                                                                                                                                                                                                                                                                                                                       +
|                                              |                                                                                                                                                                                                                                                                                  |    :width: 65                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
+                                              +                                                                                                                                                                                                                                                                                  +    :height: 21                                                                                                                                                                                                                                                                                                                                                                                                                                                       +
|                                              |                                                                                                                                                                                                                                                                                  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
+                                              +                                                                                                                                                                                                                                                                                  +                                                                                                                                                                                                                                                                                                                                                                                                                                                                      +
|                                              |                                                                                                                                                                                                                                                                                  | .. image:: images/Object_9.svg                                                                                                                                                                                                                                                                                                                                                                                                                                       |
+                                              +                                                                                                                                                                                                                                                                                  +    :width: 65                                                                                                                                                                                                                                                                                                                                                                                                                                                        +
|                                              |                                                                                                                                                                                                                                                                                  |    :height: 21                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
+                                              +                                                                                                                                                                                                                                                                                  +                                                                                                                                                                                                                                                                                                                                                                                                                                                                      +
|                                              |                                                                                                                                                                                                                                                                                  | 706 1 :math:`\begin{array}{c}{\lambda }_{S}^{T}(S)\mathrm{=}\mathrm{0,05}\mathrm{\cdot }S\\ {\lambda }_{T}^{T}(S)\mathrm{=}\mathrm{0,06}\\ {\lambda }_{\mathit{CT}}^{T}(S)\mathrm{=}\mathrm{1,539}\end{array}`                                                                                                                                                                                                                                                       |
+----------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+


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

On all edges: Zero hydraulic flow

The only driver here is the saturation of one medium by another.