Reference problem ===================== Geometry --------- The domain is an axi-symmetric slice: .. image:: images/100000000000026E0000009C168ABDF70E56E6CE.png :width: 4.8217in :height: 1.2283in .. _RefImage_100000000000026E0000009C168ABDF70E56E6CE.png: Coordinates of the points: .. csv-table:: "**Point**", ":math:`X` "," :math:`Y`" ":math:`A` ", "0.3075", "0" ":math:`B` ", "0.3375", "0" ":math:`C` ", "0.5375", "0" ":math:`D` ", "200", "0" ":math:`E` ", "200", "1" ":math:`F` ", "0.5375", "1" ":math:`G` ", "0.3375", "1" ":math:`H` ", "0.3075", "1" Area :math:`\mathrm{M1}` represents the game and is made of the material :math:`\mathrm{MATJEU}`. Zone :math:`\mathrm{M2}` represents the damaged area and is made of material :math:`\mathrm{MATZE}`. Zone :math:`\mathrm{M3}` represents the clay in the field and is made of the material :math:`\mathrm{MATCOX}`. Material properties ------------------------ Only the properties on which the solution depends are given here, knowing that the command file contains other material data that plays no role in solving the problem at hand. +------------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------+ |:math:`\mathrm{MATJEU}` | | | +------------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------+ |Liquid water |Density ( |1000 | + + :math:`{\mathrm{kg.m}}^{-3}` + + | |) | 10-3 0.5 10-9 | + + + + | | Viscosity (:math:`{\mathrm{kg.m}}^{-1}\mathrm{.}{s}^{-1}`) Compressibility (:math:`{\mathrm{Pa}}^{-1}`) | | + + + + | | | | +------------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------+ |Gaz |Density ( |2 10-3 | + + :math:`{\mathrm{kg.m}}^{-3}` + + | |) | 1.8 10-5 | + + + + | | Viscosity (:math:`{\mathrm{kg.m}}^{-1}\mathrm{.}{s}^{-1}`) | | + + + + | | | | +------------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------+ |Dissolved gas |Henry's coefficient (:math:`\mathrm{Pa.}{\mathrm{mol}}^{-1}{m}^{3}`) |1870 | +------------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------+ |Vapeur |Density (:math:`{\mathrm{kg.m}}^{-3}`) |18 10-3 | +------------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------+ |Homogenized parameters |Permeability |1.019 10-13 | + + :math:`K` + + | | ( | 0.3 0.0015. (:math:`\mathrm{Smax}-S`) 4.47 10-10. :math:`S` | + + :math:`{m}^{2}` + + | |) | | + + + + | | Porosity Fick gas (:math:`{m}^{2}\mathrm{.}{s}^{-1}`) Fick liquid (:math:`{m}^{2}\mathrm{.}{s}^{-1}`) Porosity | | + + + + | | | | +------------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------+ |Van-Genuchten parameters|:math:`N` |1.064 | + + + + | | :math:`\mathrm{Pr}` (:math:`\mathrm{Mpa}`) :math:`\mathrm{Sr}` :math:`\mathrm{Smax}` | 4.91 10-3 0.08 0.999 | + + + + | | | | +------------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------+ |Initial state |Capillary pressure (:math:`\mathrm{Mpa}`) Gas pressure (:math:`\mathrm{Mpa}`) | | + + + .. image:: images/Object_1.svg + | | | :width: 125 | + + + :height: 48 + | | | | + + + + | | | | +------------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------+ +------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------+ |:math:`\mathrm{MATZE}` | | | +------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------+ |Liquid water |Density ( |1000 | + + :math:`{\mathrm{kg.m}}^{-3}` + + | |) | 10-3 0.5 10-9 | + + + + | | Viscosity (:math:`{\mathrm{kg.m}}^{-1}\mathrm{.}{s}^{-1}`) Compressibility (:math:`{\mathrm{Pa}}^{-1}`) | | + + + + | | | | +------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------+ |Gaz |Density ( |2 10-3 | + + :math:`{\mathrm{kg.m}}^{-3}` + + | |) | 1.8 10-5 | + + + + | | Viscosity (:math:`{\mathrm{kg.m}}^{-1}\mathrm{.}{s}^{-1}`) | | + + + + | | | | +------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------+ |Dissolved gas |Henry's coefficient (:math:`\mathrm{Pa.}{\mathrm{mol}}^{-1}{m}^{3}`) |1870 | +------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------+ |Vapeur |Density (:math:`{\mathrm{kg.m}}^{-3}`) |18 10-3 | +------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------+ |Homogenized parameters |Permeability K ( |5.097 10-18 | + + :math:`{m}^{2}` + + | |) | 0.15 0.00075. (:math:`\mathrm{Smax}-S`) 2.24 10-10.S | + + + + | | Porosity Fick gas (:math:`{m}^{2}\mathrm{.}{s}^{-1}`) Fick liquid (:math:`{m}^{2}\mathrm{.}{s}^{-1}`) | | + + + + | | | | +------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------+ |Van-Genuchten parameters|:math:`N` |1.5 | + + + + | | :math:`\mathrm{Pr}` (:math:`\mathrm{Mpa}`) :math:`\mathrm{Sr}` :math:`\mathrm{Smax}` | 4.91 0 0.999 | + + + + | | | | +------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------+ |Initial state |Capillary pressure (:math:`\mathrm{Mpa}`) Gas pressure (:math:`\mathrm{Mpa}`) | | + + + .. image:: images/Object_2.svg + | | | :width: 125 | + + + :height: 48 + | | | | + + + + | | | | +------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------+ +------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------+ |:math:`\mathrm{MATCOX}` | | | +------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------+ |Liquid water |Density ( |1000 | + + :math:`{\mathrm{kg.m}}^{-3}` + + | |) | 10-3 0.5 10-9 | + + + + | | Viscosity (:math:`{\mathrm{kg.m}}^{-1}\mathrm{.}{s}^{-1}`) Compressibility (:math:`{\mathrm{Pa}}^{-1}`) | | + + + + | | | | +------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------+ |Gaz |Density ( |2 10-3 | + + :math:`{\mathrm{kg.m}}^{-3}` + + | |) | 1.8 10-5 | + + + + | | Viscosity (:math:`{\mathrm{kg.m}}^{-1}\mathrm{.}{s}^{-1}`) | | + + + + | | | | +------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------+ |Dissolved gas |Henry's coefficient (:math:`\mathrm{Pa.}{\mathrm{mol}}^{-1}{m}^{3}`) |1870 | +------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------+ |Vapeur |Density (:math:`{\mathrm{kg.m}}^{-3}`) |18 10-3 | +------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------+ |Homogenized parameters |Permeability |5.097 10-21 | + + :math:`K` + + | | ( | 0.15 0.00075. (Smax-S) 2.24 10-10.S | + + :math:`\mathrm{m2}` + + | |) | | + + + + | | Porosity Fick gas (:math:`{m}^{2}\mathrm{.}{s}^{-1}`) Fick liquid (:math:`{m}^{2}\mathrm{.}{s}^{-1}`) Porosity | | + + + + | | | | +------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------+ |Van-Genuchten parameters|:math:`N` |1.49 | + + + + | | :math:`\mathrm{Pr}` (:math:`\mathrm{Mpa}`) :math:`\mathrm{Sr}` :math:`\mathrm{Smax}` | 14.7 0.01 0.999 | + + + + | | | | +------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------+ |Initial state |Capillary pressure (:math:`\mathrm{Mpa}`) Gas pressure (:math:`\mathrm{Mpa}`) | | + + + .. image:: images/Object_3.svg + | | | :width: 125 | + + + :height: 48 + | | | | + + + + | | | | +------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------+ The saturation and permeability curves follow the Mualem-Van-Genuchten model (HYDR_VGM). It is therefore necessary to define in the materials the parameters :math:`N`, :math:`\mathrm{Pr}`,, :math:`\mathrm{Sr}`, :math:`\mathrm{Smax}`. It should be noted that these models are: .. image:: images/Object_4.svg :width: 125 :height: 48 .. _RefImage_Object_4.svg: and .. image:: images/Object_5.svg :width: 125 :height: 48 .. _RefImage_Object_5.svg: .. image:: images/Object_6.svg :width: 125 :height: 48 .. _RefImage_Object_6.svg: Relative permeability to water is expressed by integrating the prediction model proposed by Mualem (1976) into Van Genuchten's capillarity model. .. image:: images/Object_7.svg :width: 125 :height: 48 .. _RefImage_Object_7.svg: The permeability to gas is formulated in a similar way: .. image:: images/Object_8.svg :width: 125 :height: 48 .. _RefImage_Object_8.svg: We recall that for :math:`S>\mathrm{Smax}`, these curves are interpolated by a polynomial of degree 2 :math:`\mathrm{C1}` in :math:`\mathrm{Smax}`. Boundary and initial conditions ----------------------------------- A hydrogen flow and a water flow are imposed on the left border (corrosion modeling): :math:`{\mathrm{Flux}}_{\mathrm{H20}}=-\mathrm{2,13}{.10}^{-10}\mathrm{kg}/{m}^{2}s` :math:`{\mathrm{Flux}}_{\mathrm{H2}}=\mathrm{2,37}{.10}^{-11}\mathrm{kg}/{m}^{2}s` Initially, the initial capillary pressure is: 1. For game :math:`\mathrm{Pc}=\mathrm{5,18}\mathrm{Mpa}` 2. For the damaged area and the Cox: :math:`1\mathrm{atm}` The gas pressure is initially :math:`1\mathrm{atm}` everywhere.