1. Reference problem#
1.1. Geometry#
A cube with a side of \(1m\).
1.2. Material properties#
Elastic properties:
\(E+150.{10}^{6}\mathrm{Pa}\)
\(\nu =0.3\)
*Model specific settings**Infelas: *
\({\beta }_{m}=0.1142\)
Reference pressure \(A=1.\mathrm{Mpa}\)
These parameters \(A\) and \({\beta }_{m}\) were calculated in such a way as to find an inflation pressure of \(7\mathrm{MPa}\). In fact, when saturation reaches 1 (or capillary pressure 0), the swelling pressure is given by the following formula
\(\frac{{P}_{\mathrm{gf}}}{A}=\frac{\sqrt{\pi }}{2\sqrt{{\beta }_{m}}}+\frac{1}{2{\beta }_{m}}\)
knowing that the overall inflation pressure is given by the following relationship (cf. doc R7.01.41):
\({P}_{\mathit{gf}}({\mathit{Pc}}_{0})=\underset{{\mathit{Pc}}_{0}}{\overset{0}{\int }}b\left(1+\frac{\mathit{Pc}}{A}\right){e}^{-{\mathrm{\beta }}_{m}{\left(\frac{s}{A}\right)}^{2}}\mathit{dPc}\)
The shape of this pressure can be obtained by integrating and is represented by the following curve:
With these parameters, the swelling pressure stabilizes as soon as the capillary pressure exceeds 6.5 MPa.
Hydraulic properties:
Liquid water |
Density (\({\mathrm{kg.m}}^{-3}\)) Heat at constant pressure (\({\mathrm{J.K}}^{-1}\)) coefficient of thermal expansion of liquid (\({K}^{-1}\)) Compressibility (\({\mathrm{Pa}}^{-1}\)) Viscosity (\(\mathrm{Pa.s}\)) |
1.103 4180 10-4 5.10-10 10-3 |
Gas |
Molar mass (\(\mathrm{kg.}{\mathrm{Mol}}^{-1}\)) Heat at constant pressure (\({\mathrm{J.K}}^{-1}\)) Viscosity (\(\mathrm{Pa.s}\)) |
0.002 1000 9. 10-6 |
Skeleton |
Heat capacity at constant stress (\({\mathrm{J.K}}^{-1}\)) |
1000 |
Constants |
Ideal gas constant |
8,315 |
Homogenized coefficients |
Homogenized density (\({\mathrm{kg.m}}^{-3}\)) Biot coefficient Parameters of the Van-Genuchten model \(N\) \(\mathrm{Pr}(\mathrm{Mpa})\) \(\mathrm{Sr}\) |
2000 1 1.61 16.106 0 |
Reference state |
Porosity Temperature (\(°K\)) Capillary pressure (\(\mathrm{Pa}\)) Gas pressure (\(\mathrm{Pa}\)) |
0.366 303 0. 10 |
1.3. Initial conditions#
At \(t=0\):
\(\mathrm{Pgaz}=\mathrm{1atm}\)
\(S=\mathrm{0,5}\) (i.e. \(\mathrm{Pc}=\mathrm{44,7}\mathrm{Mpa}\) and \(\mathrm{pw}=-44.6\mathrm{Mpa}\))
Total stress equal to - 1atm.
1.4. Boundary conditions and loads#
Only movements BAS, DERRIERE, GAUCHE are blocked, the others are left free.
The gas pressures are set at 1 atm.
The initial saturation is \(\text{50 \%}\) (Pc = 44.7 MPa): the capillary pressure is reduced over the entire sample (which amounts to increasing the saturation, and therefore to hydration) and it is verified that at zero stress, the displacements are positive and that there is indeed swelling.
More precisely, a loading of capillary pressure decreasing linearly in \(\mathrm{1s}\) between \(\mathrm{44,7}\mathrm{Mpa}\) and \(-10\mathrm{Mpa}\) is imposed on the entire domain (cf.).
The capillary pressure becomes negative after 0.8 s. Moreover, the temperatures are constant (isothermal cases).
Figure 1.4-1: Capillary pressure loading
1.5. Bibliographical references#
Gérard, P., Charlier R., Barnichon, J.D., Su, J.D., Su, K. Shao, Su, K. Shao, J-F, Duveau, G., Giot, R., Chavant, C. Collin, F. « Numerical modeling of coupled mechanics and gas transfer » Journal of Coupled Mechanics and Gas Transfer » Journal of Theoretical and Applied Mechanics, Journal of Theoretical and Applied Mechanics, Journal of Theoretical and Applied Mechanics, Sofia, 2008, vol. 38, No. 1, pp. 101-120.