---
title: Coupled Freeflow and Porous Media Flow Models in DuMu<sup>X</sup>
---

# Coupled Freeflow and Porous Media Flow Models in DuMu<sup>X</sup>

## Coupled Freeflow and Porous Media Flow Systems?

<img src=img/FFPM-SoilWaterEvapField.png width="100%">
<img src=img/FFPM-SaltPrecip.png width="100%">
<img src=FFPM-FuelCellsSim.png width="100%">

[ETHZurich](https://emeritus.step.ethz.ch/the-step-group.html)
[EOS-SoilSalinization](https://eos.com/blog/soil-salinization/)
[EllerEtAl2011](https://iopscience.iop.org/article/10.1149/1.3596556#artAbst)

## Conceptual Physical model
<img src=img/FFPM-PhysicalModelOverview.png width="100%">

## Mathematical Model: Overview
<img src=img/FFPM-ModelConceptColumn.png width="100%">

Freeflow: NS/RANS Equations, Non-isothermal, multi-component
Porous Medium: Multi-phase Darcy, Non-isothermal, multi-component
Coupling Conditions: Local Thermodynamic Equilibrium, continuity of fluxes

## Mathematical Model: Freeflow
<img src=img/FFPM-freeflowsymbol.png width="100%">

\begin{equation}
\frac{\partial \rho_g \textbf{v}_g}{\partial t} + \nabla \cdot (\rho_g \textbf{v}_g \textbf{v}_g^T) - \DIV(\bm{\tau}_g + \bm{\tau}_{g,t})  +\DIV (p_g\textbf{I})- \rho_g \textbf{g} = 0\, .
\end{equation}

\begin{equation}
\frac{\partial \left(\rho_g X^\kappa_g\right)}{\partial t}
+ \nabla \cdot \left( \rho_g \textbf{v}_g X^\kappa_g
- \mathbf{j}_{\text{diff}}^\kappa\right)
- q^\kappa = 0\, .
\end{equation}

\begin{equation}
\frac{\partial (\rho_g u_g) }{\partial t} + \DIV (\rho_g h_g \textbf{v}_g) + \sum_{i} {\DIV (h_g^\kappa \textbf{j}_{\text{diff},t}^\kappa)} - \DIV ( (\lambda_{g} + \lambda_{t})) \grad T) = 0\, ,
\end{equation}

## Mathematical Model: Porous Medium Flow
<img src=img/FFPM-pmfsymbol.png width="100%">

\begin{equation}
	\sum\limits_{\alpha \in \{\text{l, g} \}}
	\left(\phi \frac{\partial \left(\rho_\alpha S_\alpha X_\alpha^\kappa\right)}{\partial t } + \nabla \cdot \textbf{v}_\alpha\rho_\alpha X_\alpha^\kappa + \sum_\kappa \nabla \cdot \left(  \textbf{D}_{pm,\alpha}^\kappa\rho_\alpha\nabla X_\alpha^\kappa \right)\right) = 0 ,
\end{equation}

\begin{equation}\label{eq:darcy}
	\textbf{v}_\alpha = - \frac{k_{r,\alpha}}{\mu_\alpha} K \left(\nabla p_\alpha - \rho_\alpha \textbf{g}\right) .
\end{equation}

\begin{equation}
	\sum\limits_{\alpha \in \{\text{l, g} \}}
	\left(\phi\frac{\partial \left(\rho_\alpha S_\alpha u_\alpha\right)}{\partial t}
	+ \nabla \cdot \left(\rho_\alpha h_\alpha \textbf{v}_\alpha \right)\right)
	+ \left(1- \phi\right) \frac{\partial \left(\rho_s c_{p,s}T\right)}{\partial t}
	- \nabla\cdot \left(\lambda_{pm} \nabla T \right) = 0 .
\end{equation}

## Mathematical Model: Coupling Conditions

<img src=img/FFPM-couplingsymbol.png width="100%">

\begin{equation}
[(\rho_g \textbf{v}_g) \cdot \textbf{n}]^{\text{ff}} = - [(\rho_g \textbf{v}_g + \rho_w \textbf{v}_w) \cdot \textbf{n}]^{\text{pm}}\, .
\end{equation}

<img src=img/FFPM-BJS.png width="100%">

\begin{equation}
\left[\left(- \textbf{v}_g - \frac{\sqrt{(\textbf{K}\textbf{t}_i)\cdot \textbf{t}_i}}{\alpha_{BJ}} (\nabla \textbf{v}_g + \nabla \textbf{v}_g^T)\textbf{n} \right) \cdot \textbf{t}_i \right]^{\text{ff}} = 0\, , \quad i \in \{1, .. ,\, d-1\}\, .
\end{equation}

\begin{equation}
[((\rho_g \textbf{v}_g \textbf{v}_g^T - (\bm{\tau}_g + \bm{\tau}_{g,t}) + p_g\textbf{I}) \textbf{n} )]^{\text{ff}} = [(p_g\textbf{I})\textbf{n}]^{\text{pm}}\, .
\end{equation}

\begin{equation}
[(\rho_g X_g^\kappa \textbf{v}_g + \textbf{j}_{\text{diff}, t}) \cdot \textbf{n}]^{\text{ff}} = - \left[\left( \sum_{\alpha} (\rho_{\alpha} X_{\alpha}^\kappa \textbf{v}_\alpha + \textbf{j}^\kappa_{\text{diff}, \alpha})\right) \cdot \textbf{n}\right]^{\text{pm}}\, .
\end{equation}

\begin{equation}
\left[\left(\rho_g h_g \textbf{v}_g + \sum_i h_g^\kappa \textbf{j}_{\text{diff},g}^\kappa + \lambda_{g}\grad T\right)\cdot \textbf{n}\right]^{\text{ff}} = - \left[\left( \sum_\alpha (\rho_\alpha h_\alpha \textbf{v}_\alpha + \sum_i h_\alpha^\kappa \textbf{j}_{\text{diff},\alpha}^\kappa) - \lambda_{\text{pm}}\grad T\right)\cdot \textbf{n}\right]^{\text{pm}}\, .
\end{equation}

## Numerical Model: Coupled Model
<img src=img/FFPM-numericalmodel.png width="100%">

## Soil-Water Evaporation: Further Concepts

<img src=img/FFPM-TurbulentBoundaryLayer.png width="100%">

<img src=img/FFPM-evapStages.png width="100%">
[OrEtAl2013](https://doi.org/10.2136/vzj2012.0163)

<img src=img/FFPM-evapReynoldsNum.png width="100%">

# Exercises:

## Exercise: Interface

_Tasks_

- Change flow direction for a tangetial flow as opposed to the original normal flow.
- Introduce the beavers joseph tangential flow interface condition.
- Redevelop the grid and introduce an undulating interface.
- Change the inflow boundary condition to a velocity profile.

## Exercise: Models

_Tasks_

- Modify the model to use 2phase multicomponent model in the porous medium.
- Experiment with various data output types: `.csv` and `.json`
- Visualize with various visualization tools: `gnuplot` and `matplotlib`

## Exercise: Turbulence

_Tasks_

- Introduce a Turbulence model to the free-flow domain
- Reduce the free-flow domain by using a symmetry condition at the upper domain boundary
- Vary grid resolution and perform a qualitative grid convergence test.