OGS
[case] RichardsComponentTransport

# RichardsComponentTransport process

## Governing Equations

### Richards Flow

The flow process is described by

$\phi \frac{\partial \rho_w}{\partial p} \frac{\partial p}{\partial t} S - \phi \rho_w \frac{\partial S}{\partial p_c} \frac{\partial p_c}{\partial t} - \nabla \cdot \left[\rho_w \frac{k_{\mathrm{rel}} \kappa}{\mu} \nabla \left( p + \rho_w g z \right)\right] - Q_p = 0,$

where

• $$\phi$$ is the porosity,
• $$S$$ is the saturation,
• $$p$$ is the pressure,
• $$k_{\mathrm{rel}}$$ is the relative permeability (depending on $$S$$),
• $$\kappa$$ is the specific permeability,
• $$\mu$$ is viscosity of the fluid,
• $$\rho_w$$ is the mass density of the fluid, and
• $$g$$ is the gravitational acceleration.

Here it is assumed, that

• the porosity is constant and
• the pressure of the gas phase is zero.

The capillary pressure is given by

$p_c = \frac{\rho_w g}{\alpha} \left[S_{\mathrm{eff}}^{-\frac{1}{m}} - 1\right]^{\frac{1}{n}}$

and the effective saturation by

$S_{\mathrm{eff}} = \frac{S-S_r}{S_{\max} - S_r}$

### Mass Transport

The mass transport process is described by

$\phi R \frac{\partial C}{\partial t} + \nabla \cdot \left(\vec{q} C - D \nabla C \right) + \phi R \vartheta C - Q_C = 0$

where

• $$R$$ is the retardation factor,
• $$C$$ is the concentration,
• $$\vec{q} = \frac{k_{\mathrm{rel}} \kappa}{\mu(C)} \nabla \left( p + \rho_w g z \right)$$ is the Darcy velocity,
• $$D$$ is the hydrodynamic dispersion tensor,
• $$\vartheta$$ is the decay rate.

For the hydrodynamic dispersion tensor the relation

$D = (\phi D_d + \beta_T \|\vec{q}\|) I + (\beta_L - \beta_T) \frac{\vec{q} \vec{q}^T}{\|\vec{q}\|}$

is implemented, where $$D_d$$ is the molecular diffusion coefficient, $$\beta_L$$ the longitudinal dispersivity of chemical species, and $$\beta_T$$ the transverse dispersivity of chemical species.

The implementation uses a monolithic approach, i.e., both processes are assembled within one global system of equations.

## Process Coupling

The advective term of the concentration equation is given by the Richards flow process, i.e., the concentration distribution depends on darcy velocity of the Richards flow process. On the other hand the concentration dependencies of the viscosity and density in the groundwater flow couples the unsaturated H process to the C process.

Note
At the moment there is not any coupling by source or sink terms, i.e., the coupling is implemented only by density changes due to concentration changes in the buoyance term in the groundwater flow. This coupling schema is referred to as the Boussinesq approximation.

# This process is commonly used together with the following media properties

Note: This list has been automatically extracted from OGS's benchmark tests (ctests). Therefore it might not be exhaustive, but it should give users a good overview about which properties they can/have to use with this process. Probably most of the properties occurring in this list are mandatory.

The list might contain different property <type>s for some property <name> to illustrate different possibilities the users have.