Multiscale law for water-air seepage, pollutant diffusion and advection. Inspired from WAVAT and ADVEC.

Can be parallelized with ELEMB (macroscale) or at the perturbation loop (microscale).

Takes into account the hysteresis in the water retention law when used with FKRSAT.

This law is only used for water seepage - air seepage- pollutant diffusion and advection (coupled) for non linear analysis in 2D porous media.

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

Starting from Darcy's law, the liquid water velocity is: \[ \vec{q_l} = - \frac{k_w}{\mu_w}\left[ \vec{grad}(p_w) \right]\ \text{where}\ k_w = K_w \frac{\mu_w}{\rho_w g}\left[ m^2\right] \]

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

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

Starting from Darcy's law, the gas velocity is: \[ \vec{q_g} = - \frac{k_g}{\mu_g}\left[ \vec{grad}(p_g) + \right]\ \text{où}\ k_g = K_g \frac{\mu_g}{\rho_g g}\left[ m^2\right] \]

1. Density $\rho_a$ :
   Hypothesis : The air is supposed to be a perfect gas. \[\rho_a (p_a) = \rho_{a,0}\frac{p_a}{p_{a,0}} \]
2. Intrinsic Permeability $k_g$:
   Depending on the saturation degree $S_g$ : $k_{r,g} = f(S_g)$ with $k_{g,effectif} = k_{g, intrinsic}k_{a,w}$

\[\frac{\partial}{\partial x_i} (v_i^p) = 0\]

\[ v_i^p = v_i^{advection} + v_i^{diffusion+dispersion} = C_M v_i^w - D \frac{\partial C_m}{\partial x_i} \]
With C_M and C_m [-] the concentration in pollutant at the macroscale and subscale, respectively. $v_i^w$ is the water velocity obtained from Darcy's law and $D$ [m$^2$/s] is the diffusion and dispersion coefficient.

Prepro: LHYPOFE2.F

The permeability $k_f$ is an \underline{intrinsic} permeability ($\left[L^2\right]$) $\boxed{ \begin{array}{l} k_{f, intrinsic} = K_f \frac{\mu_f}{\rho_f g}\\ \left[ L^2 \right] = \left[ LT^{-1} \right] \frac{ \left[ ML^{-1}T^{-1}\right]}{\left[ML^{-3}\right]\left[LT^{-2}\right]} \end{array}}$

Permeabilities in different directions can be input ( $I_{max}= 10$ ).The effect of these permeabilities will be summed.

Following empirical formulations for describing the evolution of the relative permeability, the thermal conductivity and saturation with the suction are possible: see Appendix 8. For any suction lower than air entry value (AIREV), the saturation is equal to SRFIELD value.
Kozeny Karman formulation: \[K = C_0 \frac{n^{EXPN}}{(1-n)^{EXPM}}\] $C_0$ is computed automatically from $C_0 = K_0 \frac{(1-n_0)^{EXPM}}{(n_0)^{EXPn}}$
GDR Momas formulation: \[ \frac{k}{k_0} = 1+EXPM\left[ \phi - \phi_0\right]^{EXPN}\ \text{où}\ EXPM = 2.10^{12}\ \text{et}\ EXPN = 3 \]

Coupling permeability-deformation formulation: (only in 2D) \[ K_{ij} = \sum_n K_n^0 (1+A\varepsilon_n^T)^3\beta_{ij} (\alpha_n) \] $\varepsilon_n^T$ tensile deformation
$\alpha_n$ crack orientation with horizontal
$A$ parameter of the crack

24

In 2D state :

In 3D state :

= 26 in 2D cases
= 16 in 3D cases