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====== ECOU-S ======

===== Description =====

Constitutive law of flow in porous media for solid elements.

==== The model ====

This law is used for non linear analysis of seepage in porous media. The case of free surface seepage is also treated. This law is used in two or three dimensional flow. \\

The mathematical model is:
  - Conservation of the mass of the fluid: \[\frac{\partial}{\partial t}(\rho_f.\theta)+div(\rho_f.\underline{q})=0\]
  - Motion of the fluid: \[\underline{q} = \frac{-k}{\mu}\left(\underline{grad}(p)+\rho_f.g.\underline{grad}(z)\right)\]


==== Files ====

Prepro: LECOUS.F \\
Lagamine: ECOU2.F, ECOU22.F, ECOU3.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| 125|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|

==== Integer parameters ====

^ Line 1 (6I5) ^^
|ISEMI| = 0 → flow analysis |
|:::| = 1 → if semi-coupled mechanical-flow analysis |
|:::| = 2 → if full coupled mechanical-flow analysis |
|IANI| = 0 → isotropic case |
|:::| ≠ 0 → anisotropic case |
|IKRN| = 0 |
|:::| = 1 → Kozeny Karman relation $K=f(n)$ |
|:::| = 2 → GDR Momas relation $K=f(n)$ |
|ISRW| Formulation index for $S_w$ (see [[appendices:a8|Appendix 8]]) |
|:::|≠ 0 → in case of seepage with free surface |
|:::| = 0 → in absence of free surface |
|IKW| Formulation index for $k_w$ (see [[appendices:a8|Appendix 8]]) |
|ISTRUCT| Formulation index for ISTRUCT |


==== Real parameters: permeability definition ====

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

__If IANI ≠ 0__
^ Line 1 (4G10.0) - Repeat IANI times (I=1,IANI) ^^
|PERMEA(I)| Soil anisotropic intrinsic permeability (k) in the direction I |
|COSX(I)| Director cosinus of the direction I |
|COSY(I)| Director cosinus of the direction I |
|COSZ(I)| Director cosinus of the direction I |

__Else, if IANI = 0__
^ Line 1 (1G10.0) ^^
|PERME| Soil isotropic intrinsic permeability (k) |

==== Real parameters ====

^ Line 1 (7G10.0) ^^
|RHO| Specific mass of the fluid $\rho_f$ [kg.m$^{-3}$] |
|POROS| Soil porosity $n_0$ |
|EMMAG| Storage coefficient $C_p$ [Pa$^{-1}$] |
|UXHIW| Fluid compressibility coefficient $1/\chi_w$ [Pa$^{-1}$] |
|POROP| Soil porosity for pollution analysis (code "TRANSPOL") |
|VISCO| Fluid dynamic viscosity $\mu_f=10^{-3}$ (default value for water at 20°C) [Pa.s] |
|PAIR| Air pressure (to define the suction for ISRW$\neq$ 0 |
^ Line 2 (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 ($S_{rfield}$) |
|AIREV| Air entry value [Pa] |
^ Line 3 (5G10.0) ^^
|CKW1| 1st coefficient of the function $k_{rw}$ |
|CKW2| 2nd coefficient of the function $k_{rw}$ |
|KRMIN| Minimum value of kr |
|CKW3| 3rd coefficient of the function $k_{rw}$ |
|CSR5| 5th coefficient of the function $S_w$ (if ISRW=26) |
^ Line 4 (8G10.0) ^^
|EXPM| km exponent for the Kozeny Karman formulation |
|EXPN| kn exponent for the Kozeny Karman formulation |
|HENRY| Henry's coefficient : solubility coefficient of air into water |
|EPSEQ0|  |
|EQ1|  |
|EQ2|  |
|EQ3|  |
|EPSEQMAX|  |
^ If ISTRUCT$\neq$0 OR ISRW=26 Line 5 (3G10.0) ^^
|SOIL MICROPOROSITY| Microstructural void ratio for dry material (if ISTRUCT$\neq$0) |
|BETA0| First coefficient for microporosity evolution (if ISTRUCT$\neq$0) |
|BETA1| Second coefficient for microporosity evolution (if ISTRUCT$\neq$0) |

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 hydraulical 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 by : 
|$\theta = n.S_r+C_p(p-CSR2)$| if ISRW=8 and p > CSR2 |
|$\theta=n.S_r+C_p.p$| if ISRW$\neq 8$ and p > 0 |

The Kozeny Karman formulation is : \[K=C_0\frac{n^{EXPN}}{(1-n)^{EXPM}}\] where $C_0$ is computed automatically from $C_0=K_0\dfrac{(1-n_0)^{EXPM}}{n_0^{EXPN}}$.\\

The GDR Momas formulation is : \[\frac{k}{k_0} = 1 + EXPM[\phi-\phi_0]^{EXPN}\] where $EXPM=2.10^{12}$ and $EXPN=3$.


===== Stresses =====

==== Number of stresses ====

5 for 3D state \\ 4 for the other cases

==== Meaning ====

For the 3-D state:

|SIG(1)| fluid mass flow in the X direction ($f_x=\rho_fq_x$) |
|SIG(2)| fluid mass flow in the Y direction ($f_y=\rho_fq_y$) |
|SIG(3)| fluid mass flow in the Z direction ($f_z=\rho_fq_z$) |
|SIG(4)| fluid mass stored as a consequence of the evolution of soil porosity ($=\rho_e=\frac{\partial}{\partial t}(\rho_f\theta)$) |
|SIG(5)| none |

For the other cases:

|SIG(1)| fluid mass flow in the X direction ($f_x=\rho_fq_x$) |
|SIG(2)| fluid mass flow in the Y direction ($f_y=\rho_fq_y$) |
|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)| none |

===== State variables =====

==== Number of state variables ====

16

==== List of state variables ====


|Q(1)| POROB*ONE*RHOWB/(RHOS*(ONE-POROB)) |
|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)| Permeability $K_{xx}$ |
|Q(7)| Permeability $K_{yy}$ |
|Q(8)| Permeability $K_{xy}$ |
|Q(10)| DLOG10($\frac{K_{xx}}{K_{xx_0}}$) |
|Q(11)| DLOG10($\frac{K_{yy}}{K_{yy_0}}$) |
|Q(13)| Microporosity |
|Q(14)| Dry density $\rho_d$ |
|Q(15)| Void ratio $e$ |
|Q(16)| VOIDMICB (mothymar law) |
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