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


Geometry
---------

We are using a one-dimensional problem in Cartesian coordinates, corresponding to a hypothesis of plane deformations in the direction :math:`y`.

The "structure" considered is finally a segment of length :math:`L\mathrm{=}20m`.


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


.. csv-table::

    "Liquid water", "Density :math:`({\mathrm{kg.m}}^{-3})`
    Specific heat at constant pressure :math:`({\mathrm{J.K}}^{-1})`
    Dynamic viscosity of liquid water :math:`(\mathrm{Pa.s})`
    Coefficient of thermal expansion of liquid :math:`({K}^{-1})`
    Compressibility :math:`({\mathit{Pa}}^{\mathrm{-}1})` "," :math:`{10}^{3}`
    :math:`4180`
    :math:`0.001`
    :math:`1.{10}^{-4}`
    :math:`{K}_{e}=5.{10}^{-10}`"
    "Solid", "Drained Young's Module :math:`E(\mathrm{Pa})`
    Poisson's ratio
    Coefficient of thermal expansion of solid :math:`({K}^{-1})` "," :math:`2.166{10}^{9}`
    :math:`0.3`
    :math:`{10}^{-5}`"
    "Initial state", "Porosity
    Temperature :math:`(K)`
    Liquid pressure :math:`(\mathrm{Pa})`
    Vapor pressure :math:`(\mathrm{Pa})` "," :math:`0.14`
    :math:`293`
    :math:`0`
    :math:`2320`"
    "Homogenized coefficients", "Homogenized density :math:`({\mathrm{kg.m}}^{-3})`
    Biot coefficient
    Intrinsic permeability :math:`({m}^{2})`
    Thermal conductivity :math:`({\mathrm{W.K}}^{-1}{m}^{-1})`
    Constant stress heat :math:`({\mathrm{J.K}}^{-1})` "," :math:`2410`
    :math:`1`
    :math:`{K}_{\text{int}}={10}^{-19}`
    :math:`{\lambda }_{T}=1.8`
    565."



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

A vertical bar in vertical positioning is heated:


.. image:: images/100002000000018900000220F00BB0564DA8722D.png
    :width: 2.1362in
    :height: 2.4661in


.. _RefImage_100002000000018900000220F00BB0564DA8722D.png:

With :math:`{\lambda }_{T}\frac{\mathrm{\partial }T}{\mathrm{\partial }x}\mathrm{=}{\Psi }_{T}` independent of time :math:`t`

Which corresponds to:


* In :math:`x\mathrm{=}0`: zero displacement, zero hydraulic flow, imposed heat flow :math:`{\Psi }_{T}=100{\mathrm{W.m}}^{-2}` constant in time


* In :math:`x\mathrm{=}L`: zero displacement, zero hydraulic flow, zero heat flow.


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

:math:`u(x)\mathrm{=}P(x)\mathrm{=}0` :math:`T(x)={T}_{0}=20°C` everywhere.