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


Geometry
---------

This one-dimensional test case comes from the literature and has an analytical solution [:ref:`1 <1>`]. It is inspired by the problem of consolidation of a porous soil saturated with water at room temperature (Terzaghi problem). Water is assumed to be incompressible (:math:`1\mathrm{/}{K}_{\mathit{lq}}\mathrm{=}\mathrm{0,0}`). The temperature is uniform. In this case, the action of gravity is neglected. The coupled saturated poro-mechanical problem is described by the displacement variables :math:`{u}_{y}` (soil compaction) and fluid pressure :math:`{P}_{\mathrm{lq}}` (the hydraulic load being :math:`{P}_{\mathrm{lq}}/{\rho }_{\mathrm{eau}}`) variables. In order to obtain a purely one-dimensional solution, the Poisson's ratio is chosen to be equal to :math:`{\nu }_{0}=\mathrm{0,0}`. The dimensions are as follows: :math:`L=\mathrm{1,00}m`, :math:`H=\mathrm{10,00}m`.

Y

F0 (t)

C

D

X

H

L
A B


.. csv-table::

    "Drained Young's Modulus: :math:`{E}_{0}=10\mathrm{MPa}` ", "Intrinsic Permeability: :math:`{K}_{\mathrm{intr}}={\mathrm{10x10}}^{-8}`"
    "Poisson's ratio: :math:`{v}_{0}=\mathrm{0,0}` ", "Fluid density: :math:`{\rho }_{\mathrm{lq}}=1000\mathrm{kg}/{m}^{3}`"
    "Density: :math:`{r}_{0}=2800\mathrm{kg}/{m}^{3}` ", "Porosity: :math:`{\phi }^{0}=\mathrm{0,5}`"
    "Biot coefficient: :math:`b=\mathrm{1,0}` ", "Dynamic viscosity of water: :math:`{\mu }_{\mathrm{lq}}(T)=1`"
    "Saturation :math:`{S}_{\mathrm{lq}}({p}_{c})=\mathrm{1,0}` ", "Relative fluid permeability: :math:`{k}_{\mathrm{lq}}^{\mathrm{rel}}({S}_{\mathrm{lq}})=1`"


Behavioral and thermal coupling characteristics are not significant.

The hydraulic permeability of the medium to water is then: :math:`{\lambda }_{\mathrm{lq}}^{H}=\frac{{K}_{\mathrm{intr}}(\varphi )\mathrm{.}{k}_{\mathrm{lq}}^{\mathrm{rel}}({S}_{\mathrm{lq}})}{{\mu }_{\mathrm{lq}}(T)}` in (:math:`{m}^{3}s/\mathrm{kg}`).

In soil mechanics, permeability is often denoted by :math:`k={\lambda }_{\mathrm{lq}}^{H}{\rho }_{\mathrm{lq}}g`, or here: :math:`k\approx {10}^{-14}m/s`.

The consolidation coefficient :math:`{c}_{\nu }={\lambda }_{\mathrm{lq}}^{H}{E}_{0}/{b}^{2}` applies here: :math:`{c}_{\nu }=\mathrm{0,1}{m}^{2}/s`.


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


Boundary conditions
~~~~~~~~~~~~~~~~~~~~~~~~~

The interstitial fluid pressure remains zero on the entire upper side :math:`\mathit{CD}`: :math:`{P}_{\mathit{lq}}=0`. The lateral faces have movements blocked in :math:`x`. The underside :math:`\mathit{AB}` has movements blocked in :math:`x` and :math:`y`, and it is waterproof: :math:`{P}_{\mathrm{lq},y}=0`


Initial conditions
~~~~~~~~~~~~~~~~~~~~~~

The column is initially at rest in a blank state: :math:`{P}_{\mathrm{lq}}=0`, :math:`{\sigma }_{\mathrm{yy}}=0`.


Loading
~~~~~~~~~~

A pressure step :math:`{F}_{0}=-\mathrm{1,0}\mathrm{Pa}` is exerted on the upper side CD to :math:`t=\mathrm{0s}`.

Gravity is neglected here.