2. Benchmark solution#

2.1. Calculation method#

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

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

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

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

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

2.2. 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:

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

  • The liquid is incompressible: \({\rho }_{l}=c\)

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

  • Relative permeability is taken to be equal to 1: \({k}_{\mathrm{rl}}=1\)

The mass conservation equation for liquid is therefore written as:

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

A really saturated model will also be tested (modeling \(D\)).

2.3. Uncertainties about the solution#

Uncertainties are zero because the reference solution is analytical.