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. Can also be used with osmotic suction (under development).

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

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

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

The water vapour only flows in unsaturated pores and depends on the tortuosity of the path: \[ \vec{i}_v = - n \; S_{r,g} \; \tau D\; \rho_s \; \vec{grad} \omega_v \] Where $\omega_v = \rho_v/\rho_g$ is the dry air mass content in the gaseous mix.

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 suction $s = p_g - p_w : S_w = f(s)$

ISR = 53 Van Genuchten model (ISR=5) with hysteresis implemented.

The main water retention curves (d=drying, w=wetting) are, according to the Van Genuchten model: \[S_{ed} = S_{res} + (S_{max}-S_{res}) \left[1 + \left(\frac{s}{a_d}\right)^{n_d}\right]^{-m_d}\] \[S_{ew} = S_{res} + (S_{max}-S_{res}) \left[1 + \left(\frac{s}{a_w}\right)^{n_w}\right]^{-m_w}\]

The hysteresis is then defined by: \[\frac{\partial S_{es}}{\partial s} (\text{wetting}) = \left(\frac{s_w}{s}\right)^b\left(\frac{\partial S_{ew}}{\partial s}\right) \text{ with } s_w = a_w \left(S_e^{-1/m_w}-1\right)^{1/n_w}\] \[\frac{\partial S_{es}}{\partial s} (\text{drying}) = \left(\frac{s_d}{s}\right)^{-b}\left(\frac{\partial S_{ed}}{\partial s}\right) \text{ with } s_d = a_d \left(S_e^{-1/m_d}-1\right)^{1/n_d}\]

And therefore: \[S_e^{t+1} = S_e^t + \left(\frac{\partial S_{es}}{\partial s}\right)\times ds\]

The ISR=53 parameters are: CSRW1=$a_d$, CSRW2=$n_d$, CSRW3=$a_w$, CSRW4=$n_w$ and CSRW5=$b$

TO BE COMPLETED.

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

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

The diffusion velocity of dry air is proportional to a density gradient. Using the diffusion theory adapted to porous medium, one writes: \[ \vec{i}_a = - n S_{r,g} \tau D \rho_g \vec{grad} \omega_a = -\vec{I}_v \] Where $\omega_a = \rho_a/\rho_g$ is the dry air mass content inside the gas mix.

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 $k_{g,effectif} = k_{g, intrinsic}k_{a,w}$
3. Saturation degree $S_g$:
   Depending on suction $s = p_g - p_w : S_g = f(s) = 1 - S_w$

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

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

Prepro: LHYPOFE2.F
Lagamine: HYPOFE2.F

To be repeated as many time as NLAWFEM2.

28

In 2D state :

11 + 5*(Number of Subscale Nodes)
/!\ The state variables vector also contains the following information for each subscale node: $X$, $Y$, $P_w$, $C$, $P_g$