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

The test case proposed by Guiducci et al.


Geometry
---------

We consider a reservoir separated into two equal parts by a horizontal discontinuity :math:`[\mathrm{EF}]`.


.. image:: images/10000000000005A1000002173D49D469963EA1D8.png
    :width: 6.211in
    :height: 2.3055in


.. _RefImage_10000000000005A1000002173D49D469963EA1D8.png:

The coordinate :math:`(x,y)` is defined by the discontinuity and the edge of the structure. We note :math:`\alpha` the angle between the :math:`(x,y)` coordinate system and the global :math:`(X,Y)` coordinate system.

Coordinates of the points (in meters) in coordinate system :math:`(x,y)`:


.. csv-table::

    "", ":math:`x` "," :math:`y` ", "", ":math:`x` "," :math:`y`"
    ":math:`A` ", "0", "-150"," -150", ":math:`D` ", "0", "150"
    ":math:`B` ", "2500", "-150"," -150", ":math:`E` ", "0", "0"
    ":math:`C` ", "2500", "0"," 0", ":math:`F` "," ", "2500", "0"



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


* Properties of intersticial fluid (liquid water):


.. csv-table::

    "Density", ":math:`1000{\mathrm{kg.m}}^{-3}`"
    "Viscosity", ":math:`{1.10}^{-3}\mathrm{Pa.s}`"
    "Compressibility", ":math:`{3.10}^{9}\mathrm{Pa}`"



* Properties of the rock matrix:

The matrix is elastic and has the following properties:


.. csv-table::

    "Young's module", ":math:`200\mathrm{MPa}`"
    "Poisson's Ratio", ":math:`\mathrm{0,25}`"
    "Porosity", ":math:`\mathrm{0,4055}`"
    "Intrinsic permeability", ":math:`\mathrm{1,688}{.10}^{-17}{m}^{2}`"



* Properties of the discontinuity:

The laws of discontinuity behavior are detailed in the documentation [:ref:`R7.02.15 <R7.02.15>`].

The mechanical behavior of the discontinuity is given by Bandis's law.

Note :math:`\varepsilon` the crack opening, :math:`{U}_{\text{max}}` the asymptrotic opening under zero stress, and :math:`U={U}_{\text{max}}-\varepsilon` the crack closure.

In the direction normal to the crack, we have


.. math::
 :label: eq-1

    d\ sigma {\ text {'}} _ {n} =- {n} =- {K} _ {\ text {ni}}\ frac {\ mathrm {dU}} {{(1-\ frac {U} {U} {{U}} _ {\ mathrm {max}}})} ^ {\ gamma}}

In the tangential direction, we have


.. math::
 :label: eq-1

    \ sigma {\ text {'}} _ {t} = {k} = {K} _ {t} {u} _ {t}


.. csv-table::

    "Initial normal stiffness :math:`{K}_{\text{ni}}` "," :math:`{1.10}^{9}{\mathrm{Pa.m}}^{-1}`"
    "Asymptotic opening :math:`{U}_{\mathrm{max}}` "," :math:`\mathrm{2,17}\mathrm{mm}`"
    "Coefficient :math:`\gamma` "," :math:`2`"
    "Tangential stiffness :math:`{K}_{\text{t}}` "," :math:`{1.10}^{12}{\mathrm{Pa.m}}^{-1}`"


The flow in the crack is given by the cubic law.


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

The initial conditions are as follows:


* initial opening :math:`{\varepsilon }_{0}`: :math:`\mathrm{3,04}{.10}^{-4}m`


* initial pressure in the massif: :math:`\mathrm{48,7}\mathrm{MPa}`


* compressive stress in both directions of the plane: :math:`62\mathrm{MPa}`


Boundary conditions
----------------------

The mechanical and hydraulic boundary conditions are given by figures to.

On :math:`[\mathrm{AB}]`: movements blocked in :math:`y` and zero hydraulic flow

On :math:`[\mathrm{BC}]`: movements blocked in :math:`x` and zero hydraulic flow

On :math:`[\mathrm{CD}]`: mechanical pressure of :math:`62\mathrm{MPa}` and zero hydraulic flow

On :math:`[\mathrm{DA}]`: mechanical pressure of :math:`62\mathrm{MPa}` and imposed pressure :math:`{p}^{\text{*}}`

Note: mechanical pressure is only applied to the solid (not to the crack)

The evolution of the pressure :math:`{p}^{\text{*}}` imposed by the well located in :math:`x=0` is as follows (see also figure):


* Linear decrease from :math:`\mathrm{48,7}\mathrm{MPa}` to :math:`\mathrm{33,7}\mathrm{MPa}` during the first :math:`\mathrm{7,5}` years of operation


* Maintains at :math:`\mathrm{33,7}\mathrm{MPa}` for the following :math:`\mathrm{12,5}` years.


.. image:: images/100000000000043D000000FB0CE3B7F04F728249.jpg
    :width: 6.3008in
    :height: 1.4571in


.. _RefImage_100000000000043D000000FB0CE3B7F04F728249.jpg:

Illustration 1.4.1: Boundary conditions and mechanical initials


.. image:: images/1000000000000363000000A32A63D4773B4FC88B.jpg
    :width: 5.8575in
    :height: 1.1008in


.. _RefImage_1000000000000363000000A32A63D4773B4FC88B.jpg:

Illustration 1.4.2: Boundary conditions and hydraulic initials


.. image:: images/1000000000000305000002655FD28B4D4B2DD3B3.jpg
    :width: 2.8472in
    :height: 2.2575in


.. _RefImage_1000000000000305000002655FD28B4D4B2DD3B3.jpg:

Figure 1.4.3: Operating scenario