====== INTEC2 ======
===== Description =====
Constitutive law of longitudinal and transversal flow in porous media for a interface element ([[elements:fail2|FAIL2B]] or [[elements:fain2|FAIN2B]])
==== The model ====
This law is only  used for non linear analysis of longitudinal seepage in porous media 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
    * INDIC = 2 if the normal to the structure intersects one segment, this segment pressure is chosen; otherwise, no flux is computed (interest if 2 layers of contact element exist)

==== Files ====
Prepro: LINTEC.F \\
Lagamine: INTEC2.F
===== Availability =====
|Plane stress state| YES |
|Plane strain state| YES |
|Axisymmetric state| YES |
|3D state| NO |
|Generalized plane state| YES |
===== Input file =====
==== Parameters defining the type of constitutive law ====
^ Line 1 (2I5, 60A1)^^
|IL|Law number|
|ITYPE| 117 |
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|
==== Integer parameters ====
^ Line 1 (4I5) ^^
|INDIC|= 0, 1, 2 to define the outside pressure used in case of no contact (see [[laws:intec2#The model|The model]])|
|IKE|index of the longitudinal permeability formulation :|
|:::| = 0 → $k_l = k_{l0}$.|
|:::| = 1 → $k_l = f(d) = \frac{\left(D_0 + V\right)^{exp}}{12} = \frac{d^{exp}}{12}$.|
|ITR|index of transmissivity:|
|:::| = 0 if [[elements:fail2|FAIL2]] element;|
|:::| = 1 if [[elements:fain2|FAIN2]] element.|
|IDDL|DDL number (3 = water, 4 = air, 5 = temperature), only for the case NTANA=5 and FAIL2 element. If NTANA $\neq$ 5 or [[elements:fain2|FAIN2]] element, IDDL is always equal to 3 (Default value).|

==== Real parameters ====
The longitudinal permeability $k$ is an __intrinsic__ permeability $\left(\left[L^2\right]\right)$ \\
($K_l$ is the permeability coefficient $(\left[LT^{-1}\right])$ \[ k_{f,intrinsic} = K_l \frac{\mu_f}{\rho_f g} \\ \left[ L^2 \right] = \left[ LT^{-1} \right] \frac{ \left[ ML^{-1}T^{-1}\right]}{\left[ML^{-3}\right]\left[LT^{-2}\right]} \]
^ Line 1 (7G10.0) ^^
|PERMEA|fault longitudinal intrinsic permeability (=$k_{l0}$)|
|RHO|specific mass of the fluid (=$\rho_f$)|
|POROS|fault porosity (=$n_0$)|
|EMMAG|storage coefficient (=$C_p$)|
|ALPHA|$\alpha$ *  parameter used to|	
|BETA|$\beta$ *  define the curve $\theta = \theta(p)$|
|VISCO|fluid dynamic viscosity ($\mu_f=10^{-3}$=default value for water at 20°C)|
^ Line 2 (6G10.0) ^^
|THCON|fault transverse transmissivity ($T_{t\_c}$) when contact occurs|
|CONVEC|fault transverse transmissivitty ($T_{t\_nc}$) when contact does not occur|
|PAMB|atmosphere pressure|
|D0|maximal fault closure in absolute value (correspond to D0 from [[laws:intme|INTME2]] mechanical law) for formulation (IKE=1)|		
|EXP|exponent (=$exp$) = 2 for cubic law|			
|EPAIS|fault thickness (useful only if no Goodman's formulation in mechanical law)|

The longitudinal permeability of the fault is computed according to IKE value : 
  * IKE = 0 : $ k_{long} $ = PERMEA
  * IKE = 1 : $ k_{long} = (D0 + V)^{EXP} = d^{EXP} $ \\ where $V$ is the fault closure computed by the mechanical law and $d = D0+V$ represents the actual fault opening. \\ In this case, the [[laws:intme|INTME2]] mechanical parameters $(D0, \gamma, K_n\  \text{and}\ \sigma’)$ are linked with the hydraulic parameter $k_{long}$: \[
\begin{array}{l}
V = D_0\left[ \sqrt[1-\gamma]{\left( \frac{(1-\gamma)}{D_0K_n}\sigma’ + 1\right)}-1\right] \\
d = \sqrt{12 k_{long}} \\ d-V = D_0 \end{array}\] The knowledge of four parameters is sufficient to determine the fifth parameter.
The evolution of the stored fluid volume ($\theta$) with the fluid pressure ($p$) is given by the following functions:
  * in case of seepage with free surface: \[
\theta = n_0 \left( \frac{1}{\pi} \arctan\left(\frac{p-\beta}{\alpha}\right) +\frac{1}{2}\right) + C_p^* \left(p-\beta\right)\] with $C_P^* = C_p$ below the free surface \\ and $C_P^* = 0$ above the free surface
  * in absence of free surface: \[n = n_0 + C_p\]

===== Stresses =====
==== Number of stresses ====
= 3 : if [[elements:fail2|FAIL2]] element \\ 
= 4 : if [[elements:fain2|FAIN2D]] element
==== Meaning ====
|SIG(1)|longitudinal flow in the interface element|
|SIG(2)|fluid flow stored as a consequence of the evolution of soil porosity|
|SIG(3)|1st transversal fluid flow in the interface element|
|SIG(4)|2nd transversal fluid flow in the interface element (only if [[elements:fain2|FAIN2]])|


===== State variables =====
==== Number of state variables ====
4
==== List of state variables ====
|Q(1)|for [[elements:fain2|FAIN2]] element: pore pressure inside the fault |
|:::|for [[elements:fail2|FAIL2]] element: 0|
|Q(2)|intrinsic longitudinal permeability $(=k_{long})$ |
|Q(3)|transverse transmissivity $(T_{t\_c}\ \text{or}\ T_{t\_nc})$ |