====== HMIC ======
===== Description =====
2D hydraulic microscopic law for solid elements.\\
Can be parallelised in ELEMB (at the macro-scale) or in the perturbation loop (at the micro-scale).\\ \\

The law definition and typical values of parameters for clays can be found in Corman (2024)((Corman, G. (2024). Hydro-mechanical modelling of gas transport processes in clay host rocks in the context of a nuclear waste repository. PhD thesis, University of Liège. https://hdl.handle.net/2268/307996)).


==== The model ====
This law replaces the macroscopic fluid law, by considering a complete hydraulic microstructure, made of dominant horizontal bedding planes, vertical bridging planes and matrix blocks. Under the assumption of the spatial repeticion of the microstructure over the distance $w$, a Representative Element Volume (REV) is built, including fractures and tubes whose behaviours are governed by constitutive laws. Fluid pressures and fluxes are computed at the microscopic scale in that hydraulic network. This way, the law is used for water seepage, air seepage, diffusion and advection (coupled) under non-linear analysis in 2D porous media. Effects of mechanics on the flow are implicitely integrated into the microscale model by means of hydro-mechanical couplings.
{{ :laws:hmic.png?600 |}}
=== Mass balance equation for water ===
\[
\underbrace{\frac{\partial}{\partial t} (\rho_s . n . S_{r,w}) + div(\rho_w \vec{q_l})}_{\text{Liquid water}} 
 = 0
\]

=== Liquid water flow ===
From Darcy's law, the advective component of the liquid water flow respectively reads for a fracture and a tube:
\[
\vec{q_l} = - \frac{k_{r_w}}{\mu_w}\frac{1}{A}\kappa\left[ \vec{grad}(p_w) + g \rho_w \vec{grad}(y)\right]\]
where\\
\[ 
\kappa 
= {}
\begin{cases}
-\frac{h_b^2}{12}h_b \cdot w, fracture\\
-\pi \frac{D^4}{128}, tube\\
\end{cases}
\]
\[
k_{r_w}
=
\begin{cases}
\frac{S_{r}^{*^2}}{2}(3-S_{r}^{*}), fracture\\
S_{r}^{*^2}, tube
\end{cases}
\]

=== Liquid state equations ===

  - Density $\rho_w$: \[\rho_w (T, p_w) = \rho_{wo}\left[ 1+\frac{p_w-p_{w0}}{\chi_w} \right]\]
  - Intrinsic permeability $k_w$: \\ Depending on the water saturation degree $S_w$ : $k_{r,w} = f(S_w)$ avec $k_{w,eff} = k_f k_{r,w}$
  - Saturation degree $S_w$: \\ Depending on succion $s = p_a - p_w : S_w = f(s)$

=== Mass balance equation for air ===
\[\frac{\partial}{\partial t} (\rho_a . n . S_{r,g}) + div(\rho_a \vec{q_g}) + div(\vec{i_a}) = 0\]

=== Dry air flow ===
From Darcy's law, the advective component of the liquid water flow respectively reads for a fracture and a tube:
\[
\vec{q_g} = - \frac{k_{r_g}}{\mu_g}\frac{1}{A}\kappa\left[ \vec{grad}(p_g) + g \rho_g \vec{grad}(y)\right]\]
where\\
\[ 
\kappa 
= {}
\begin{cases}
-\frac{h_b^2}{12}h_b \cdot w, fracture\\
-\pi \frac{D^4}{128}, tube\\
\end{cases}
\]
\[
k_{r_g}
=
\begin{cases}
(1-S_{r}^*)^3, fracture\\
(1-S_{r}^*)^2, tube
\end{cases}
\]


From Fick's law, the diffusive component of the dissolved air flow respectively reads for a fracture and a tube:
\[
\vec{i}_a = - n S_{r,g} \tau D \rho_g \vec{grad} (\omega_a) 
\]


where $\omega_a = \rho_a/\rho_g$.

=== Dry gas state equations ===

  - Density $\rho_a$ :\\ //Assumption of classical ideal gas equation of state: \[\rho_a (T, p_a) = \rho_{a,0}\frac{p_a}{p_{a,0}}\frac{T_0}{T} \]//
  - Perméabilité intrinsèque $k_g$: \\ Depending on the saturation degree $S_g$ : $k_{r,g} = f(S_g)$ avec $k_{g,effectif} = k_{g, intrinsic}k_{a,w}$
  - Gaseous saturation degree $S_g$: \\ Depending on suction $s = p_g - p_w$ \\ $S_g = 1-S_w$ 


==== Files ====
Prepro: LHMIC.F & EHMICA.F \\
Lagamine: HMIC.F & EHMICB.F \\
===== Availability =====
|Plane stress state| NO |
|Plane strain state| YES |
|Axisymmetric state| YES |
|3D state| YES |
|Generalized plane state| NO |
===== Input file =====
==== Parameters defining the type of constitutive law ====
^ Line 1 (2I5, 60A1)^^
|IL|Law number|
|ITYPE| 628 |
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|
==== Integer parameters ====
^ Line 1 (4I10) ^^
|NLAWFEM2|Number of constitutive laws at the sub-scale|
|KFLU|Number of DOF: 1=Pw, 2=Pw+Pg|
|IGAS|Type of gas: 0=Air, 1=H2, 2=N2, 3=Ar, 4=He, 5=CO2, 6=CH4|
|IDIFF|Activation of diffusion mechanism: 0=No, 1=Yes,  |
^ Line 2 (1G10.0) ^^
|FACONV|Units of conversion of the REV (it has a size of 1 [-])|

==== Real parameters ====
^ Line 1 (5G10.0) ^^
|VISCW0|Liquid dynamic viscosity $(=\mu_{w,0})\ \left[ Pa.s \right]$|
|RHOW0|Liquid density $(=\rho_{w,0})\ \left[ kg.m^{-3}\right]$|
|UXHIW|Liquid compressibility coefficient $(=1/ \chi_{w})\ \left[ Pa^{-1}\right]$|
|PW0|Initial water pressure $\left[ Pa\right]$|
|T0|Initial temperature $\left[ K\right]$|
^ Line 2 (3E10.2,2G10.0) ^^
|VISCA0|Gas dynamic viscosity $(=\mu_{a,0})\ \left[Pa.s \right]$|
|RHOA0|Gaz density $(=\rho_{a,0})\ \left[kg.m^{-3}\right]$|
|PMGAS|Gas molar mass $[g/mol]$|
|PA0|Initial gas pressure $\left[ Pa\right]$|
|PHENRY|Henry coefficient $\left[ -\right]$|

==== Sub-scale parameters ====
To be repeated as many time as NLAWFEM2.
^ Line 1 (7I5) ^^
|ILAW2|No. of the sub-scale constitutive law (=1:NLAWFEM2)|
|ITYPE2|Type of sub-scale law: 1=Fracture (manual), 2=Fracture (automatic), 3=Tube (manual), 4=Tube (automatic), 5=Bridge (manual), 6=Bridge (automatic)|
|ISR|Retention curve: 1=Brooks-Corey for fracture, 2=Brooks-Corey for tube, 3=van Genuchten for fracture, 4=van Genuchten for tube|
|IKW|Water relative permeability curve |
|IKA|Gas relative permeability curve|
|INUMEL2|Number of micro-elements with this law|
|ICONST|Constant element opening: 0=No, 1=Yes|
^ Line 2 - Retention curve coefficients (4G10.0) ^^
|PE0|Initial air entry pressure of the micro-element|
|CDF|Exponent parameter|
|SRES|Residual saturation degree $(=S_{res})$|
|SRG0|Initial gas saturation|
|AKRMIN|Minimum value of relative permeability|
|SRFIELD|Field saturation degree $(=S_{r, field})$|
|CDF2|Exponent parameter|
|CSR8|8th parameter of ISR|
^ Line 3 - Fracture law coefficients (4G10.0) ^^
|AKP|Stiffness parameter of the material|
|GAMMA|Exponent parameter|
|DINI|Initial aperture|
|DMAX|Maximum aperture|
^ Line 3 - Tube law coefficients (3G10.0) ^^
|DINI|Initial aperture|
|DMAX|Maximum aperture|
|TORT|Tortuosity|

===== Stresses =====
==== Number of stresses ====
28
==== Meaning ====
__In 2D state :__
|SIG(1)|$\sigma_x$ (unused)|
|SIG(2)|$\sigma_y$ (unused)|
|SIG(3)|$\sigma_{xy}$ (unused)|
|SIG(4)|$\sigma_z$ (unused)|
|SIG(5)|Homogenised liquid flow along $x$ $(=f_{wx})$|
|SIG(6)|Homogenised liquid flow along $y$ $(=f_{wy})$|
|SIG(7)|Homogenised liquid flow stored $(=f_{we})$|
|SIG(8)|Homogenised gas flow along $x$ $(=f_{ax})$|gas advection + \\ gas diffusion + \\ dissolved gas advection + \\ dissolved gas diffusion|
|SIG(9)|Homogenised gas flow along $y$ $(=f_{ay})$|:::|
|SIG(10)|Homogenised gas flow stored $(=f_{ae})$|:::|
|SIG(11)|Advection dissolved gas flow along $x$ $(=f_{ad,x})$|
|SIG(12)|Advection dissolved gas flow along $y$ $(=f_{ad,y})$|
|SIG(13)|Diffusion dissolved gas flow along $x$ $(=f_{add,x})$|
|SIG(14)|Diffusion dissolved gas flow along $y$ $(=f_{add,y})$|
|SIG(15)|Advection gaseous gas flux along $x$ $(=f_{ag,x})$|
|SIG(16)|Advection gaseous gas flux along $y$ $(=f_{ag,y})$|
|SIG(18)|Unused|
|SIG(18)|Unused|
|SIG(19)|Unused|
|SIG(20)|Unused|
|SIG(21)|Unused|
|SIG(22)|Unused|
|SIG(23)|Unused|
|SIG(24)|Unused|
|SIG(25)|Unused|
|SIG(26)|Unused|
|SIG(27)|Unused|
|SIG(28)|Unused|

===== State variables =====
==== Number of state variables ====
=6 in 2D cases
==== List of state variables ====
|Q(1)|Unused|
|Q(2)|Unused|
|Q(3)|Homogenised macro-scale porosity|
|Q(4)|Homogenised macro-scale saturation|
|Q(5)|Water storage|
|Q(6)|Gas storage|
|Q(7)|Saved fracture aperture of the current step (from 7 to 7+nico)|
|Q(8)|Unused|
|Q(9)|Unused|
|Q(10)|Unused|
|Q(11)|Unused|
|Q(12)|Unused|