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).

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.

\[ \underbrace{\frac{\partial}{\partial t} (\rho_s . n . S_{r,w}) + div(\rho_w \vec{q_l})}_{\text{Liquid water}} = 0 \]

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
\[ \kappa = {} \begin{cases} -\frac{h_b^2}{12}h_b \cdot w, fracture\\ -\pi \frac{D^4}{128}, tube\\ \end{cases} \] \[ k_{r_w} = \begin{cases} \frac{S_{r}^{*^2}}{2}(3-S_{r}^{*}), fracture\\ S_{r}^{*^2}, tube \end{cases} \]

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

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

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 $\omega_a = \rho_a/\rho_g$.

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

Prepro: LHMIC.F & EHMICA.F
Lagamine: HMIC.F & EHMICB.F

To be repeated as many time as NLAWFEM2.

28

In 2D state :

10 + 5*(Number of Subscale Nodes)
/!\ The state variables vector also contains the following information for each subscale node: X,Y,Pw,C,Pg