====== PTRNS ======
===== Description =====
Advection‑diffusion constitutive law in saturated-unsaturated porous media for solid eulerian elements (element [[elements:trpo2|TRPO2]]) \\
(**P**ollutant **TR**ansport in **N**on **S**aturated conditions)

==== The model ====
This law is only used for linear pollutant transport in isotropic solids by upwind methods in saturated-non saturated porous media. \\
This constitutive law takes into account the advection‑dispersion in the moving fluid but also degradation, adsorption on the solid matrix and immobile fluid effect as linear phenomena. This law is used for two‑dimensional flow.

==== Files ====
Prepro: LPTRNS.F \\
===== Availability =====
|Plane stress state| YES |
|Plane strain state| YES |
|Axisymmetric state| YES |
|3D state| NO |
|Generalized plane state| NO |
===== Input file =====
==== Parameters defining the type of constitutive law ====
^ Line 1 (2I5, 60A1)^^
|IL|Law number|
|ITYPE| 131 |
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|
==== Integer parameters ====
^ Line 1 (3I5) ^^
|INDV|= 0 constant advection|
|:::|= 1 space time variation of advection|
|:::|= 2 only space variation of advection|
|:::|= 3 semi coupled analysis with flow|
|INDM|= 0 same meshing for flow and pollution|
|:::|= 1 different meshing|
|INDS|= 0 fully saturated porous media|
|:::|= 1 saturated-unsaturated porous media|
^Line 2 - Only if INDS = 1 (9I5) ^^
|IFAT|law formulation indexes for the 9 parameters \\ (respectively $a_T, a_L, D_m, R_{dm}, R_{dim}, A_m, A_{im}, \alpha_m, \alpha_{im}$) \\ in unsaturated condition: \\ 0 = constant formulation \\ 1 = linear formulation|
|IFAL|:::|
|IFDM|:::|
|IFROM|:::|
|IFRDIM|:::|
|IFAM|:::|
|IFAIM|:::|
|IFALM|:::|
|IFALIM|:::|

==== Real parameters ====
^ Line 1 (7G10.0)^^
|UABS|- constant apparent velocity modulus|
|TETA|constant apparent velocity direction (angle in RAD)|
|CIM0|initial immobile water concentration|
|ALALX|first facultative imposed upwind in X direction|
|ALALY|first facultative imposed upwind in Y direction|
|BEBEX|second facultative imposed upwind in X direction|
|BEBEY|second facultative imposed upwind in Y direction|
^Line 2 (7G10.0)^^
|ATRANS|transversal dispersivity of the porous medium $(a_T)$ in|
|ALONG|longitudinal dispersivity of the porous medium $(a_L)$ fully|
|DIFFM|	molecular diffusion of the porous medium $(D_m)$ saturated|
|RDM|retardation factor for mobile water $(R_{dm})$ conditions|
|RDIM|retardation factor for immobile water $(R_{dim})$|
|AM|degradation constant for mobile water $(A_{m})$ |
|AIM|degradation constant for immobile water $(A_{im})$ |
^Line 3 (3G10.0)^^
|ALM|transfer constant for mobile water $(\alpha_{m})$ |
|ALIM|transfer constant for immobile water $(\alpha_{im})$|
|PEFF|effective surface porosity of the porous medium.|

__Only if INDS = 1:__
^Line 4 (3G10.0)^^
|CAT1|1st coefficient of the function $a_T(S_r)$|
|CAT2|2nd coefficient of the function $a_T(S_r)$|
|CAT3|3rd coefficient of the function $a_T(S_r)$|
^Line 5 (3G10.0)^^
|CAL1|1st coefficient of the function $a_L(S_r)$|
|CAL2|2nd coefficient of the function $a_L(S_r)$|
|CAL3|3rd coefficient of the function $a_L(S_r)$|
^Line 6 (3G10.0)^^
|CDM1|1st coefficient of the function $D_m(S_r)$|
|CDM2|2nd coefficient of the function $D_m(S_r)$|
|CDM3|3rd coefficient of the function $D_m(S_r)$|
^Line 7 (3G10.0)^^
|CRDM1|1st coefficient of the function $R_{dm}(S_r)$|
|CRDM2|2nd coefficient of the function $R_{dm}(S_r)$|
|CRDM3|3rd coefficient of the function $R_{dm}(S_r)$|
^Line 8 (3G10.0)^^
|CRDIM1|1st coefficient of the function $R_{dim}(S_r)$|
|CRDIM2|2nd coefficient of the function $R_{dim}(S_r)$|
|CRDIM3|3rd coefficient of the function $R_{dçm}(S_r)$|
^Line 9 (3G10.0)^^
|CAM1|1st coefficient of the function $A_{m}(S_r)$|
|CAM2|2nd coefficient of the function $A_{m}(S_r)$|
|CAM3|3rd coefficient of the function $A_{m}(S_r)$|
^Line 10 (3G10.0)^^
|CAIM1|1st coefficient of the function $A_{im}(S_r)$|
|CAIM2|2nd coefficient of the function $A_{im}(S_r)$|
|CAIM3|3rd coefficient of the function $A_{im}(S_r)$|
^Line 11 (3G10.0)^^
|CALM1|1st coefficient of the function $\alpha_{m}(S_r)$|
|CALM2|2nd coefficient of the function $\alpha_{m}(S_r)$|
|CALM3|3rd coefficient of the function $\alpha_{m}(S_r)$|
^Line 12 (3G10.0)^^
|CALIM1|1st coefficient of the function $\alpha_{im}(S_r)$|
|CALIM2|2nd coefficient of the function $\alpha_{im}(S_r)$|
|CALIM3|3rd coefficient of the function $\alpha_{im}(S_r)$|

__Remarks on parameters__
  - Upwind parameters:
    * If ALALX and ALALY $\in \left[0,1\right] \rightarrow$ HUGHES formulation for the first upwind \\ BEBEY and BEBEY $\in \left[0,1\right] \rightarrow$ HUGHES formulation for the first upwind
    * If ALALX $\in \left[0,1\right] \rightarrow$ formulation in FRENET axes for the first upwind
    * If BEBEX $\in \left[0,1\right] \rightarrow$ formulation in FRENET axes for the second upwind
    * Otherwise optimum formulation in FRENET axes
  - Physical parameters:
    * RDM = $1 + \theta_s \rho_s p K_d / \theta_m$
    * RDIM = $1 + \theta_s \rho_s (1-p) K_d / \theta_{im}$
    * AM = $k_s.(RDM -1).+ k_m  + ALM$
    * AIM = $k_s.(RDIM -1).+ k_{im} + ALIM$
    * ALM = $\alpha_\alpha/\theta_m$
    * ALIM = $\alpha_\alpha/\theta_{im}$ \\ With: 
      * $\theta_s$ = 1-total porosity
      * $\theta_{m}$ = effective porosity
      * $\theta_{im}$ = non effective porosity
      * $k_{s,m,im}$ = degradation coefficients in solid, moving and non-moving fluids
      * $K_{d}$ = linear adsorption coefficient
      * $p$ = part of solid surface in contact with the moving fluid
      * $\rho_s$ = solid density
      * $\alpha_d$ = exchange coefficient between mobile and immobile fluids
Following empirical formulations for describing the evolution of the transport parameters "P" ($=a_T, a_L$; $D_m, R_{dm}, R_{dim}, A_m, A_{im}, \alpha_m, \alpha_{im}$) with the saturation $S_r$ are possible ($P_{sat}$ = value of the parameter in fully saturated conditions, when $S_r = 1$) :

  * constant formulation : $P(S_r) = P_{sat}$
  * linear formulation : $P(S_r) = P_{sat}\left[ (1-coeff.1).S_r + coeff.1\right] \Rightarrow coeff1 = \frac{P(S_r = 0)}{P_{sat}}$

===== Stresses =====
==== Number of stresses ====
8
==== Meaning (numerical, not physical) ====
|SIG(1)|pollutant flow in the X direction $(=q_X)$|
|SIG(2)|pollutant flow in the Y direction $(=q_Y)$|
|SIG(3)|pollutant dispersive flux in X direction|
|SIG(4)|pollutant dispersive flux in Y direction|
|SIG(5)|pollutant storage due to time variation of concentration|
|SIG(6)|pollutant storage by convection and not restored by degradation|
|SIG(7)|pollutant flux from moving to non moving fluid|
|SIG(8)|pollutant degradation|

===== State variables =====
==== Number of state variables ====
6
==== List of state variables ====
|Q(1)| fluid Darcy velocity in the X direction |
|Q(2)| fluid Darcy velocity in the Y direction |
|Q(3)| fluid storage |
|Q(4)| immobile water concentration. |
|Q(5)| mobile water concentration (if INDV = 3) |
|Q(6)| saturation degree |