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


Geometry
---------

We consider a cylinder with radius :math:`1\mathit{cm}` and height :math:`1\mathit{cm}` (i.e. a mesh corresponding to a square domain of :math:`1\mathit{cm}\mathrm{\times }\mathrm{1cm}`, the modeling being axisymmetric).


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

Here, parameters corresponding to argillite are selected in order to obtain a realistic thermal pressurization coefficient.


.. 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})` (if constant, see next section)
    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:`\mathrm{3,14}{10}^{9}`
    :math:`0.375`
    :math:`{10}^{-5}`"
    "Reference state", "Porosity
    Temperature :math:`(K)`
    Liquid pressure :math:`(\mathrm{Pa})` "," :math:`0.18`
    :math:`273`
    :math:`0`"
    "Homogenized coefficients", "Homogenized density :math:`({\mathrm{kg.m}}^{-3})`
    Biot coefficient
    Intrinsic permeability :math:`({m}^{2})`
    Thermal conductivity", ":math:`2410`
    :math:`0.6`
    :math:`{K}_{\text{int}}={10}^{-21}`
    :math:`{\lambda }_{T}=1.61`"



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


.. image:: images/1000020100000211000001748F2E4FBB6F1EF336.png
    :width: 1.2528in
    :height: 1.0791in


.. _RefImage_1000020100000211000001748F2E4FBB6F1EF336.png:

We impose:


* On the bottom and left edges: zero displacements, zero hydraulic flow, zero heat flow. These are symmetry conditions.


* On the top and right edges: Total stress imposed on :math:`12\mathit{MPa}`, zero hydraulic flow, temperature imposed as a function of time :math:`T(t)` following a linear ramp such as:

:math:`T(t)={T}_{0}+\frac{\Delta T}{{t}_{s}}` where :math:`{t}_{s}` corresponds to the simulation time (here :math:`{t}_{\mathit{sim}}\mathrm{=}\mathrm{1h}`) and :math:`\Delta T` the temperature variation imposed during this time (here :math:`\Delta T\mathrm{=}40°C`).


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

:math:`P(x)\mathrm{=}\mathrm{4MPa}` and :math:`T(x)\mathrm{=}{T}_{0}\mathrm{=}20°C` everywhere.