1. Reference problem#

1.1. Description#

The test consists in desaturating a permeable porous sample with boundary conditions that are initially impermeable at the edges (zero outflow of water). At moment \(t=0\), an « opening » of 50cm is imposed on the right edge using the oozing conditions, as illustrated below. The water level decreases progressively until it stabilizes at the level of the lower edge of the opening.

_images/10000201000003D4000002C4BE6B87806D1344A7.png

Image 1.1-a: Test case description

1.2. Geometry#

The dimensions of the rectangular sample are:

  • height: \(h=3m\)

  • width: \(l\mathrm{=}1m\)

The length of the ooze section is: :math:`e=50\mathit{cm}`. It is placed on the right edge, between a height of 1m and 1.5m.

1.3. Material properties#

The elastic properties are:

  • \(E=515\mathit{MPa}\)

  • \(\mathrm{\nu }=0.3\)

  • \(\mathrm{\rho }=2670\mathit{kg}\mathrm{.}{m}^{-3}\)

The law of behavior associated with hydraulic model HH2M is LIQU_GAZ: we therefore consider 1 liquid phase without dissolved air and a gas phase without water vapor.

The properties of the hydraulic model are shown in the table below:

Liquid water

Density (\(\mathit{kg}\mathrm{.}{m}^{-3}\)) Inverse of the compression coefficient (\({\mathit{Pa}}^{\mathrm{-}1}\)) Intrinsic permeability (\({m}^{2}\)) viscosity (\(\mathit{Pa}\mathrm{.}s\))

103 0 10-9 10-3

Gas

Molar mass (\(\mathit{kg}\mathrm{.}{\mathit{mole}}^{-1}\))

29,965.10-3

Initial state

Porosity Temperature (\(K\)) Capillary pressure (\(\mathit{Pa}\)) Gas pressure (\(\mathit{Pa}\)) Initial liquid saturation

0.4 293 0 1 0.9999

Constants

Ideal gas constant

8,315

Homogenized coefficients

Homogenized density (\({\mathit{kg.m}}^{\mathrm{-}3}\)) Biot coefficient Capillary curve

2670 1 \({S}_{r}\left({P}_{c}\right)={\left(1+{\left(\frac{{P}_{c}}{{10}^{+7}}\right)}^{1.7}\right)}^{-\mathrm{0,412}}\)

Table 1.3-1: Hydraulic Properties

1.4. Boundary conditions and loads#

The boundary conditions are:

  • Blocking the vertical movement of the base: \(\mathit{DY}=0\)

  • Blocking horizontal movement on both edges: \(\mathit{DX}=0\)

  • Zero water and air pressures on the top edge: \(\mathit{PRE}1=\mathit{PRE}2=0\)

  • Seep conditions on the « opening » section: \(\mathit{PRE}1\ge 0\)

The imposed load is the acceleration of gravity.

The calculation is done between \(t=0\) and \(t=\mathrm{0,2}s\) in steps from \(\mathrm{0,001}s\) to \(t=\mathrm{0,02}s\), then in steps from \(\mathrm{0,01}s\) to \(t=\mathrm{0,2}s\).

1.5. Initial conditions#

The initial effective stress in the sample is isotropic and geostatic, i.e. equal to:

\(\mathrm{\sigma }{\text{'}}_{\mathit{XX}}=\mathrm{\sigma }{\text{'}}_{\mathit{YY}}=\mathrm{\sigma }{\text{'}}_{\mathit{ZZ}}=(\mathrm{\rho }-{10}^{3})gz\)

The initial water pressure is geostatic: \(\{\begin{array}{c}\mathit{PRE}1={10}^{3}gz\\ \mathit{PRE}2=0\end{array}\)