====== INTEC3 ======
===== Description =====
Constitutive law of longitudinal  flow in porous media for a 3d interface element ([[elements:fail3|FAIL3B]])

==== The model ====
This law is only  used for non linear analysis of longitudinal seepage in porous media 3d interface element.\\
The case of free surface seepage is also treated.\\
Transversal fluid transfer between the bodies depends upon the contact state. 
  - Contact occurs (pression non zero) fluid transfer is computed according the transverse transmissivity $T_{t\_c}$.
  - Contact does not occur, fluid transfer is computed by convection with transverse transmissivity $T_{t\_nc}$. \\ In this case, the outside pressure is the following one:
    * INDIC = 1 always the atmosphere pressure
    * INDIC = 0 if the normal to the structure intersects one segment, this segment pressure is chosen; otherwise, the atmosphere pressure is used

=== Mathematical model ===

  - Conservation of the mass of the fluid: \[ \frac{\partial}{\partial t}(\rho_f \theta) + div (\rho_f \vec{q}) = 0\]
  - Motion of the fluid:\[\vec{q} = \frac{-k}{\mu} \left( \vec{grad}(p)+\rho_f g \vec{grad}(z)\right)\]
==== Files ====
Prepro: LINTEC3.F \\
Lagamine: INTEC3.F
===== Availability =====
|Plane stress state| NO |
|Plane strain state| NO |
|Axisymmetric state| NO |
|3D state| YES |
|Generalized plane state| NO |
===== Input file =====
==== Parameters defining the type of constitutive law ====
^ Line 1 (2I5, 60A1)^^
|IL|Law number|
|ITYPE| 118 |
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|
==== Integer parameters ====
^ Line 1 (7I5) ^^
|IANI|= 0 → isotropic case|
|:::|$\neq$ 0 → anisotropic case|
|ISRW|formulation index for $S_W$ : (see [[appendices:a8|Appendix 8]]) \\ in case of seepage with free surface: ISRW $\neq$ 0 \\ in absence of free surface : ISRW = 0|
|IKW|formulation index for $k_W$ (see [[appendices:a8|Appendix 8]])|
|INDIC|= 0 or 1 to define the outside pressure used in case of no contact (see [[laws:intec3#The model|The model]])|
|IKE|index of the longitudinal permeability formulation:|
|:::|= 0 → $k_l = k_{l0}$.|	
|:::|= 1 → $k_l = f(d) = \left(\frac{D_0 + V}{12}\right)^{exp}$.|
|ITR|index of transmissivity: __not used now__|
|:::|= 0, if [[elements:fail3|FAIL3]] element|
|:::|= 1, if [[elements:fain3|FAIN3]] element|
|IDDL|DDL number (4 = water, 5 = air, 6 = temperature), only for the case NTANA=13 and [[elements:fail3|FAIL3]] element. If NTANA $\neq$ 13 or [[elements:fain3|FAIN3]] element, IDDL is always equal to 4 (Default value).|

==== Real parameters: permeability definition ====
The permeability $k$ is an __intrinsic__ permeability $\left(\left[L^2\right]\right)$\\
($K$ is the permeability coefficient $(\left[LT^{-1}\right])$ \[ k_{intrinsic} = K \frac{\mu_f}{\rho_f g}\\ \left[ L^2 \right] = \left[ LT^{-1} \right] \frac{ \left[ ML^{-1}T^{-1}\right]}{\left[ML^{-3}\right]\left[LT^{-2}\right]} \]\\

^If IANI $\neq$ 0, then for I = 1, IANI (3G10.0) ^^
|PERMEA(I)|soil anisotropic int. permeability (k) in the direction I|
|COSX(I)|director cosinus of the direction I|
|COSY(I)|director cosinus of the direction I|
^Else (IANI = 0) (1G10.0)^^
|PERME|soil isotropic int. permeability (k)|


==== Real parameters : permeability definition ====
^ Line 1 (6G10.0/7G10.0/7G10.0/4G10.0/2G10.0 ) ^^
|D0|asymptotic fault opening (=$d_0$) for formulation (IKE=1):|		
|EXP|exponent (=$exp$) =  2  for cubic law|			
|THCON|fault transverse transmissivity ($T_{t\_c}$) when contact occurs|
|CONVEC|fault transverse transmissivity ($T_{t\_nc}$) when contact does not occur|
|PAMB|atmosphere pressure|
|EPAIS|fault thickness (useful only if no Goodman's formulation in mechanical law)|
^ Line 2 (7G10.0) ^^
|RHO|specific mass of the fluid $(=\rho_f)\ \left[kg m^{-3}\right]$|
|POROS|soil porosity (=$n_0$)|
|EMMAG|storage coefficient $(=C_p)\ \left[Pa^{-1}\right]$|
|UXHIW|fluid compressibility coefficient $1/\chi_W$ if ISEMI = 1 or 2 $\left[Pa^{-1}\right]$|
|POROP|soil porosity for pollution analysis (code "TRANSPOL »)|
|VISCO|fluid dynamic viscosity ($=\mu_f = 10^{-3}$=default value for water at 20°C) $\left[Pa s\right]$|
|PAIR|Air pressure ( To define the suction for ISRW $\neq$ 0 )|
^ Line 3 (7G10.0) ^^
|CSR1|1st  coefficient of the function $S_W$|
|CSR2|2nd coefficient of the function $S_W$|
|CSR3|3rd coefficient of the function $S_W$|
|CSR4|4th coefficient of the function $S_W$|
|SRES|residual saturation degree ( = $S_{res}$ )|
|SRFIELD|field saturation degree ( = $S_{rfiled}$)|
|AIREV|air entry value $\left[Pa\right]$|
^ Line 1 (4G10.0) ^^
|CKW1|1st  coefficient of the function $k_{rw}$|
|CKW2|2nd coefficient of the function $k_{rw}$|
|KRMIN|Minimum value of $k_{r}$|
|CKW3|3rd coefficient of the function $k_{rw}$|
^ Line 1 (2G10.0) ^^
|HENRY|Henry’s coefficient : solubility coefficient of air into water|

Following empirical formulations for describing the evolution of the relative permeability, and saturation with the suction are possible : see [[appendices:a8|Appendix 8]] \\
\\
The storage coefficient $C_p$ allows to take into account the variation of the water stored in the pore due to soil deformations in a hydraulic analysis. \\
The volume of water stored is given by the following relation:
\[ \theta = n S_r \]
In order to take soil deformations into account, the volume of water stored is given be:
\[ \theta = n S_r + C_p (p-CSR2)  \text{ if ISRW } = 8 \text{ and } p>CSR2 \\
\theta = n S_r + C_p p \text{ if ISRW } \neq 8 \text{ and } p>0 \]
The longitudinal permeability of the fault is computed according to IKE value :  
  * IKE = 0 : $k_{long}$ = PERMEA
  * IKE = 1 : $k_{long} = (D0 + )^{EXP}$ where $V$ is the fault closure computed by the mechanical law and $(D0+V)$ represents the actual fault opening.

===== Stresses =====
==== Number of stresses ====
4
==== Meaning ====
|SIG(1)|fluid mass flow in the x local direction $(f_x = \rho_f q_x$|
|SIG(2)|fluid mass flow in the y local direction $(f_y = \rho_f q_y$|
|SIG(3)|fluid mass stored as a consequence of the evolution of soil porosity $(=\rho_e = \frac{\partial}{\partial t}(\rho_f \theta))$|
|SIG(4)|transversal fluid flow though the interface element|

===== State variables =====
==== Number of state variables ====
6
==== List of state variables ====
|Q(1)|For [[elements:fain3|FAIN3]] element: pore pressure inside the fault. __Not used now__|
|:::|For [[elements:fail3|FAIL3]] element: 0|
|Q(2)|Soil  isotropic permeability ($= k$)|
|Q(3)|Soil porosity ($= n_0$)|
|Q(4)|Saturation (only with free surface): \\ $S_r = 1$ if $p \geq  0$ \\ $S_r = \frac{\theta}{n_0}$ if $p < 0$|
|Q(5)|actualised fluid specific mass|
|Q(6)|transverse transmissivity ($T_{t\_c}$ or $T_{t\_nc}$)|