Coupled Free-Flow and Porous Media Flow Models in DuMux

Motivation

Environmental and Agricultural Issues

Fig.1 - Evaporation of soil water (Heck et al. (2020))1
  • Evaporation of soil water
  • Soil salinization
  • Underground storage (e.g. CO2, atomic waste)

Technical Issues

Fig.2 - Filter (Schneider et al. (2023))2
  • Fuel cells
  • Filters (e.g. air)
  • Heat exchangers (e.g. CPU cooling)

Biological Issues

Fig.3 - Brain tissue (Koch et al. (2020))3
  • Brain tissue
  • Leaf structure

Model Overview

Conceptual Physical Model

Fig.4 - Coupled dynamics at the soil-atmosphere interface (Photo: Edward Coltman)

Conceptual Physical Model

Fig.5 - Exchange processes at the free-flow porous-medium interface at different scales (Photo: Martin Schneider)

Mathematical Model: Overview

Free Flow:

  • Stokes / Navier-Stokes / RANS
  • 1-phase, n-components, non-isothermal

Interface conditions:

  • no thickness, no storage
  • local thermodynamic equilibrium
  • continuity of fluxes
  • continuity of state variables

Porous media:

  • Darcy / Forchheimer
  • m-phases, n-components, non-isothermal

Mathematical Model: Free Flow

Mathematical Model: Free Flow

  • Momentum balance \[ \frac{\partial \rho_g \textbf{v}_g}{\partial t} + \nabla \cdot (\rho_g \textbf{v}_g \textbf{v}_g^T) - \nabla \cdot (\mathbf{\tau}_g + \mathbf{\tau}_{g,t}) +\nabla \cdot (p_g\textbf{I})- \rho_g \textbf{g} = 0 \]

  • Component mass balance \[ \frac{\partial \left(\rho_g X^\kappa_g\right)}{\partial t} + \nabla \cdot \left( \rho_g X^\kappa_g \textbf{v}_g + \mathbf{j}_{\text{diff}}^\kappa\right) - q^\kappa = 0 \]

  • Energy balance \[ \frac{\partial (\rho_g u_g) }{\partial t} + \nabla \cdot (\rho_g h_g \textbf{v}_g) + \sum_{\kappa} {\nabla \cdot (h_g^\kappa \textbf{j}_{\text{diff},t}^\kappa)} - \nabla \cdot ( (\lambda_{g} + \lambda_{t}) \nabla T) = 0 \]

Mathematical Model: Porous Medium Flow

Mathematical Model: Porous Medium Flow

  • Darcy velocity (momentum balance) \[ \textbf{v}_\alpha = - \frac{k_{r,\alpha}}{\mu_\alpha} K \left(\nabla p_\alpha - \rho_\alpha \textbf{g}\right) \]

  • Component mass balance \[ \sum\limits_{\alpha \in \{\text{l, g} \}} \left(\phi \frac{\partial \left(\rho_\alpha S_\alpha X_\alpha^\kappa\right)}{\partial t } + \nabla \cdot \rho_\alpha X_\alpha^\kappa \textbf{v}_\alpha + \nabla \cdot \mathbf{j}_{\text{diff}}^\kappa\right) = 0 \]

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

Mathematical Model: Coupling Conditions

  • Total mass condition \[ [(\rho_g \textbf{v}_g) \cdot \textbf{n}]^{\text{ff}} = - [(\rho_g \textbf{v}_g + \rho_w \textbf{v}_w) \cdot \textbf{n}]^{\text{pm}} \]

Mathematical Model: Coupling Conditions

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

Mathematical Model: Coupling Conditions

  • Momentum (normal) condition \[ [((\rho_g \textbf{v}_g \textbf{v}_g^T - (\mathbf{\tau}_g + \mathbf{\tau}_{g,t}) + p_g\textbf{I}) \textbf{n} )]^{\text{ff}} = [(p_g\textbf{I})\textbf{n}]^{\text{pm}}\, \]

  • Component mass condition \[ [(\rho_g X_g^\kappa \textbf{v}_g + \textbf{j}_{\text{diff}^\kappa}) \cdot \textbf{n}]^{\text{ff}} = - \left[\left( \sum_{\alpha} (\rho_{\alpha} X_{\alpha}^\kappa \textbf{v}_\alpha + \textbf{j}^\kappa_{\text{diff}, \alpha})\right) \cdot \textbf{n}\right]^{\text{pm}}\, \]

  • Energy condition \[ \left[\left(\rho_g h_g \textbf{v}_g + \sum_i h_g^\kappa \textbf{j}_{\text{diff},g}^\kappa - (\lambda_{g} + \lambda_{t})\nabla T\right)\cdot \textbf{n}\right]^{\text{ff}} = - \left[\left( \sum_\alpha (\rho_\alpha h_\alpha \textbf{v}_\alpha + \sum_\kappa h_\alpha^\kappa \textbf{j}_{\text{diff},\alpha}^\kappa) - \lambda_{\text{pm}}\nabla T\right)\cdot \textbf{n}\right]^{\text{pm}}\, \]

Numerical Model: Coupled Model

Fig.6 - Discretization scheme (Fetzer (2018))4

Example: Soil-Water Evaporation

Soil-Water Evaporation: Soil-Water Evaporation

Example: Soil-Water Evaporation

Fig.7 - Evaporation in the water cycle (Shahraeeni et al. (2012))5

Example: Soil-Water Evaporation

Fig.8 - Different evaporation stages (Or et al.(2013))6

Example: Simple Evaporation Setup

Tab1: Input parameter

Parameter Value
\(\textbf{v}_g^{ff}\) [m/s] (3.5,0)\(^T\)
\(p_g^{ff}\) [Pa] 1e5
\(X_g^{w,ff}\) [-] 0.008
\(T^{ff}\) [K] 298.15
\(p_g^{pm}\) [Pa] 1e5
\(S_l^{pm}\) [-] 0.98
\(T^{pm}\) [K] 298.15

Fig.9 - Model setup (Fetzer (2018))4

Example: Results

Fig.10 - Results: Evaporation from a simple setup (Fetzer (2018))4

Exercises

Exercise: Interface

Tasks

  • Change flow direction for a tangential 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 a 2-phase 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

References

  1. Heck, K., Coltman, E., Schneider, J. and Helmig, R. (2020). Influence of radiation on evaporation rates: A numerical analysis. Water Resources Research, 56, e2020WR027332. https://doi.org/10.1029/2020WR027332

  2. Schneider, M., Gläser, D., Weishaupt, K., Coltman, E., Flemisch, B. and Helmig, R. (2023). Coupling staggered-grid and vertex-centered finite-volume methods for coupled porous-medium free-flow problems. Journal of Computational Physics. 112042. https://doi.org/10.1016/j.jcp.2023.112042.

  3. Koch, T., Flemisch, B., Helmig, R., Wiest, R. and Obrist, D. (2020). A multiscale subvoxel perfusion model to estimate diffusive capillary wall conductivity in multiple sclerosis lesions from perfusion MRI data. Int J Numer Meth Biomed Engng. 36:e3298. https://doi.org/10.1002/cnm.

  4. Fetzer, Thomas: Coupled Free and Porous-Medium Flow Processes Affected by Turbulence and Roughness – Models, Concepts and Analysis, Universität Stuttgart. - Stuttgart: Institut für Wasser- und Umweltsystemmodellierung, 2018

  5. Shahraeeni, E., Lehmann, P. and Or, D. (2012). Coupling of evaporative fluxes from drying porous surfaces with air boundary layer: Characteristics of evaporation from discrete pores. Water Resources Research. 48. 9525-. 10.1029/2012WR011857.

  6. Or, D., Lehmann, P., Shahraeeni, E. and Shokri, N. (2013), Advances in Soil Evaporation Physics—A Review. Vadose Zone Journal, 12: 1-16 vzj2012.0163. https://doi.org/10.2136/vzj2012.0163