====== FMIVP ======
===== Description =====
Constitutive law for mixed limit condition for element [[elements:fmivp|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) ^^
|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 $S_{r,w}^f$).

{{  :laws:evaporation_flow_seepage_flow.png?400  |}}
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.