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FMIVP
Description
Constitutive law for mixed limit condition for element FMIVP (seepage and evaporation)
The model
This law is only used for non linear analysis of solids. This constitutive law allows to impose a mixed limit condition on a boundary, with a classical penalty method, combining with an evaporation boundary condition.
Files
Prepro: LFMIVP.F
Availability
| Plane stress state | YES |
| Plane strain state | YES |
| Axisymmetric state | YES |
| 3D state | YES |
| Generalized plane state | YES |
Input file
Parameters defining the type of constitutive law
| Line 1 (2I5, 60A1) | |
|---|---|
| IL | Law number |
| ITYPE | 198 |
| COMMENT | Any comment (up to 60 characters) that will be reproduced on the output listing |
Integer parameters
| Line 1 (2I5) | |
|---|---|
| IDDL | DDL number (3 = water, 4 = air, 5 = temperature, in 2D case) |
| DDL number (4 = water, 5 = air, 6 = temperature, in 3D case) | |
| ISRW | Formulation index for retention curve $S_{rw}$ |
Real parameters
| Line 1 (5G10.0/6G10.0 ) | |
|---|---|
| COEFK | K penalty coefficient |
| ALPHA | Mass transfer coefficient $\left[m/s\right]$ |
| PG0 | Definition gas pression $\left[Pa\right]$ |
| T0 | Definition temperature pression $\left[ K\right]$ |
| L | Latent heat of the liquid $\left[ J/kg\right]$ |
| BETA | Convective heat transfer coefficient |
| Line 2 (6G10.0) | |
| CSR1 | 1st coefficient of the function $S_{rw}$ |
| CSR2 | 2nd coefficient of the function $S_{rw}$ |
| CSR3 | 3rd coefficient of the function $S_{rw}$ |
| CSR4 | 4th coefficient of the function $S_{rw}$ |
| SRES | residual saturation degree ( = $S_{res}$) |
| SRFIELD | field saturation degree ( = $S_{r,field}$) |
| AIREV | air entry value $\left[Pa\right]$ |
Stresses
Number of stresses
3
Meaning
| SIG(1) | =water output or input flow at the boundary |
| SIG(2) | =gas output or input flow at the boundary |
| SIG(3) | =temperature output or input flow at the boundary |
State variables
Number of state variables
5
List of state variables
| Q(1) | = 0 |
| Q(2) | = porous surface relative humidity |
| Q(3) | = drying air relative humidity |
| Q(4) | = total water flow |
| Q(5) | = evaporation flow |
The total water flow boundary condition is expressed as the sum of the seepage flow and vapour exchange flow
\[
\vec{q} = \vec{S} + \vec{E}
\]
A ramp function gives the expression of the seepage liquid flow $\vec{S}$ :
\[
\left\{
\begin{array}{l}
\vec{S} = K_{pen} . (p_w^f - p_{atm})^2\ \text{if}\ p_w^f \geq p_w^{cav}\ \text{and}\ p_w^f \geq p_{atm}\\
\vec{S} = 0 \ \text{if}\ p_w^f < p_w^{cav}\ \text{or}\ p_w^f < p_{atm}
\end{array}
\right.
\]
With $p_w^f$ the pore water pressure in the rock mass formation, $p_w^{cav}$ the water pressure corresponding to the relative humidity in the cavity (using Eq. 10), $p_{atm}$ the atmospheric pressure and $k_{pen}$ a seepage transfer coefficient.
The evaporation exchange is expressed as the difference of vapour density between the tunnel atmosphere and rock mass: \[ \vec{E} = \alpha_0 S_{r,w}^f (\rho_v^f - \rho_v^{cav}) \] With $\rho_v^f$ and $\rho_v^{cav}$ vapour density respectively in the formation and in the cavity and $\alpha$ a vapour mass transfer coefficient (that depends on the degree of saturation $S_{r,w}^f$).
De la même manière, on exprime que l’évaporation en surface dépend des conditions thermiques. Le flux de chaleur $\vec{t}$ de la frontière vers l’extérieur est exprimé par :
\[
\vec{t} = L \vec{q} - \beta \left( T_{air} - T_{roche}^\Gamma \right)
\]
Avec $T_{air}$ et $T_{roche}^\Gamma$ la température respectivement de l’air ambiant et en paroi d’échantillon, $\beta$ un coefficient de transfert de chaleur et $L$ la chaleur latente de vaporisation (= 2500 kJ/kg). Le premier terme correspond à l’énergie consommée pour la vaporisation de l’eau en paroi, tandis que le second terme correspond au flux de chaleur convectif entre l’atmosphère et le milieu poreux.
