Benchmark solution
=====================


Calculation method
-----------------

Or a bar saturated with water (considered to be incompressible). This bar of length :math:`L` and height :math:`h` has an initial pressure :math:`{P}_{l}(x,y,t=0)={P}_{i}` and is subjected to a pressure gradient such as :math:`{P}_{l}(\mathrm{0,}y,t)={P}_{G}` and :math:`{P}_{l}(L,y,t)={P}_{i}`.

This evolution problem leads after a time :math:`{t}_{p}` to a linear permanent state such as :math:`P(x,y,t>{t}_{p})=\frac{{P}_{\mathrm{ini}}-{P}_{G}}{L}x+{P}_{G}`

The flow of water :math:`{M}_{11}` (pressure gradient factor) is then constant along the bar. If we integrate it on a vertical section :math:`\Gamma` of the bar, and with an outgoing normal :math:`\nu`, we get:

:math:`{\mathrm{\int }}_{\Gamma }{M}_{11}\mathrm{.}\nu \mathrm{=}\mathit{h.}{\rho }_{l}\frac{{K}_{\mathit{int}}}{{\mu }_{l}}\mathrm{.}\frac{{P}_{G}\mathrm{-}{P}_{\mathit{ini}}}{L}\mathrm{.}x\mathrm{.}\nu`

The calculation of this integral will be carried out in this test case on 3 surfaces (or side in :math:`\mathrm{2D}`).


Simplifying hypotheses
---------------------------

In order to test the calculation of flows on the most complete hydraulic model possible, we start from a two-phase modeling that is degenerated into monophasic modeling. To do this, it is considered that the medium is completely saturated with water and a zero gas pressure is imposed on all the nodes. The biphasic system then comes down to solving the following problem:

:math:`\frac{\mathrm{\partial }(\varphi {\rho }_{l})}{\mathrm{\partial }t}\mathrm{-}\mathit{div}({K}_{\mathit{int}}\frac{{\rho }_{l}{k}_{\mathit{rl}}}{{\mu }_{l}}\mathrm{\nabla }{P}_{l})\mathrm{=}0`


* The liquid is incompressible: :math:`{\rho }_{l}=c`


* The matrix is compressible and the porosity evolves in proportion to the liquid pressure: :math:`\frac{\partial \varphi }{\partial {P}_{l}}={E}_{m}`


* Relative permeability is taken to be equal to 1: :math:`{k}_{\mathrm{rl}}=1`

The mass conservation equation for liquid is therefore written as:

:math:`{\rho }_{l}{E}_{m}\frac{\partial {P}_{l}}{\partial t}-d({K}_{i}\frac{{\rho }_{l}}{{\mu }_{l}}\nabla {P}_{l})=0`

A really saturated model will also be tested (modeling :math:`D`).


Uncertainties about the solution
----------------------------

Uncertainties are zero because the reference solution is analytical.