---
title: Introduction to DuMu^x^
subtitle: Overview and Available Models
---

# Table of Contents

## Table of Contents

1. [Structure and Development History](#structure-and-development-history)
2. [Mathematical Models](#available-models)
3. [Spatial Discretization](#spatial-discretization)
4. [Model Components](#model-components)
5. [Simulation Flow](#simulation-flow)

# Structure and Development History

## DuMu^x^ is a DUNE module

<img src="img/dumux_dune_module.png"/>

## The DUNE Framework

* **Developed** by scientists at around 10 European research institutions.
* **Separation** of data structures and algorithms by abstract interfaces.
* Efficient implementation using **generic** programming techniques.
* **Reuse** of existing FE packages with a large body of functionality.
* Current stable release: **2.9** (November 2022).

## DUNE Core Modules

* **dune-common:** basic classes
* **dune-geometry:** geometric entities
* **dune-grid:** abstract grid/mesh interface
* **dune-istl:** iterative solver template library
* **dune-localfunctions:** finite element shape functions

## Overview

<img src="img/dumux.png" width="300"/>

* **DuMu^x^:** DUNE for Multi-{Phase, Component, Scale, Physics, $\text{...}$} flow and transport in porous media.
* **Goal:** **sustainable, consistent, research-friendly framework** for the implementation and application of
**FV discretization schemes**, **model concepts**, and **constitutive relations**.
* **Developed** by more than 30 PhD students and post docs, mostly at LH^2^ (Uni Stuttgart).

## Applications

<div style="text-align: left">
**Successfully applied** to
</div>
* gas (CO~2~, H~2~, CH~4~, $\text{...}$) storage scenarios
* environmental remediation problems
* transport of substances in biological tissue
* subsurface-atmosphere coupling (Navier-Stokes / Darcy)
* flow and transport in fractured porous media
* root-soil interaction
* pore-network modelling
* developing new finite volume schemes

## DuMu^x^ Modules

* [**dumux-lecture**](https://git.iws.uni-stuttgart.de/dumux-repositories/dumux-lecture): example applications for lectures offered by LH^2^, Uni Stuttgart
* [**dumux-pub/---**](https://git.iws.uni-stuttgart.de/dumux-pub): code and data accompanying a publication (reproduce and archive results)
* [**dumux-appl/---**](https://git.iws.uni-stuttgart.de/dumux-appl): Various application modules (many not publicly available, e.g. ongoing research)

## Development History

* 01/2007: Development **starts**.
* 07/2009: Release **1.0**.
* 09/2010: **Split** into stable and development part.
* 12/2010: Anonymous **read access** to the **SVN** trunk of the stable part.
* 02/2011: Release **2.0,** $\text{...}$, 10/2017: Release **2.12**.
* 09/2015: Transition from Subversion to **Git**.
* 12/2018: Release **3.0,** $\text{...}$, 07/2024: Release **3.9**.

## Funding

Efforts mainly funded through ressources at the LH^2^: [Department of Hydromechanics and Modelling of Hydrosystems at the University of Stuttgart](https://www.iws.uni-stuttgart.de/en/lh2/)
and third-party funding aquired at the LH^2^

<img src="img/lh2.jpeg" width="300"/>

## Funding

We acknowledge funding that supported the development of DuMu^x^ in past and present:

<img src="img/funding.svg" width="550"/>

## Downloads and Publications

* More than 1000 unique release **downloads**.
* More than 250 peer-reviewed [**publications**](https://puma.ub.uni-stuttgart.de/group/dumux/dumuxarticle?resourcetype=publication&items=1000&sortPage=year) and [PhD theses](https://puma.ub.uni-stuttgart.de/group/dumux/dumuxphd?resourcetype=publication&items=1000&sortPage=year) using DuMu^x^.

## Evolution of C++ Files

<img src="img/files_vs_releases.png" width="600"/>

## Evolution of Code Lines

<img src="img/lines_vs_releases.png" width="600"/>

## Folder structure

```code
dumux
├── build-cmake
│   ├── examples
│   ├── test
│   ├── ...
├── doc
├── dumux
├── examples
├── test
└── ...
```
* Further details can be found in the corresponding [documentation](https://dumux.org/docs/doxygen/master/directory-structure.html)

## Mailing Lists and GitLab

* **Mailing lists** of DUNE (<dune@dune-project.org>) and DuMu^x^ (<dumux@listserv.uni-stuttgart.de>)
* GitLab **Issue Tracker** (<https://git.iws.uni-stuttgart.de/dumux-repositories/dumux/issues>)
* Get **GitLab** accounts (non-anonymous) for better access
    * DUNE GitLab (<https://gitlab.dune-project.org/core>)
    * DuMu^x^ GitLab (<https://git.iws.uni-stuttgart.de/dumux-repositories/dumux>)

## Documentation

* Code **documentation**
    * DuMu^x^ (<https://dumux.org/docs/doxygen/master/>)
    * DUNE (<https://dune-project.org/doxygen/>)
* DuMu^x^ **Examples** (<https://git.iws.uni-stuttgart.de/dumux-repositories/dumux/-/tree/master/examples#examples>)
* DuMu^x^ **Website** (<https://dumux.org/>)

# Mathematical Models

## Mathematical Models

Preimplemented models:

* **Flow in porous media (Darcy)**: Single and multi-phase models for flow and transport in porous materials.
* **Free flow (Navier-Stokes)**: Single-phase models based on the Navier-Stokes equations.
* **Shallow water flow**: Two-dimensional shallow water flow (depth-averaged).
* **Geomechanics**: Models taking into account solid deformation of porous materials.
* **Pore network**: Single and multi-phase models for flow and transport in pore networks.

## Flow in Porous Media

<img src="img/models.png" width="650"/>

## Darcy's law

* Describes the advective flux in porous media on the macro-scale

* Single-phase flow
$$
\mathbf{v} = - \frac{\mathbf{K}}{\mu} \left(\textbf{grad}\, p - \varrho \mathbf{g} \right)
$$

* Multi-phase flow (phase $\alpha$)
$$
\mathbf{v}_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left(\textbf{grad}\, p_\alpha - \varrho_\alpha \mathbf{g} \right)
$$
where $k_{r\alpha}(S_\alpha)$ is the relative permeability, a function of saturation $S_\alpha$.

* For non-creeping flow, Forchheimer's law is available as an alternative.

## 1p -- Single-Phase

* Uses standard Darcy approach for the conservation of momentum by default
* Mass continuity equation
$$
\frac{\partial\left( \phi \varrho \right)}{\partial t} + \text{div} \left( \varrho \mathbf{v} \right) = q
$$

* Primary variable: $p$

* Further details can be found in the corresponding [documentation](https://dumux.org/docs/doxygen/master/group___one_p_model.html)

## 1pnc -- Single-Phase, Multi-Component

* Uses standard Darcy approach for the conservation of momentum by default
* Transport of component $\kappa \in \{\text{H2O}, \text{Air}, ...\}$
$$
\frac{\partial\left( \phi \varrho X^\kappa \right)}{\partial t} + \text{div} \left( \varrho X^\kappa \mathbf{v} - \varrho D^\kappa_\text{pm} \textbf{grad} X^\kappa \right) = q
$$

* Closure relation: $\sum_\kappa X^\kappa = 1$
* Primary variables: $p$ and $X^\kappa$

* Further details can be found in the corresponding [documentation](https://dumux.org/docs/doxygen/master/group___one_p_n_c_model.html)

## 1pncmin -- with Fluid-Solid Phase Change

* Transport equation for each component $\kappa \in \{\text{H2O}, \text{Air}, ...\}$
$$
\frac{\partial \left( \varrho_f X^\kappa \phi \right)}{\partial t} + \text{div} \left( \varrho_f X^\kappa \mathbf{v} - \mathbf{D_\text{pm}^\kappa} \varrho_f \textbf{grad}\, X^\kappa \right) = q_\kappa
$$

* Mass balance solid phases
$$
\frac{\partial \left(\varrho_\lambda \phi_\lambda \right)}{\partial t} = q_\lambda
$$

* Primary variables: $p$, $X^k$ and $\phi_\lambda$

* Further details can be found in the corresponding [documentation](https://dumux.org/docs/doxygen/master/group___one_p_n_c_min_model.html)

## 2p -- Two-Phase Immiscible

* Uses standard multi-phase Darcy approach for the conservation of momentum by default
* Conservation of the phase mass of phase $\alpha \in \{\text{w}, \text{n}\}$
$$
\frac{\partial \left(\phi \varrho_\alpha S_\alpha \right)}{\partial t} + \text{div} \left(\varrho_\alpha \mathbf{v}_\alpha \right)  = q_\alpha  
$$

* Constitutive relations: $p_\text{c} := p_\text{n} - p_\text{w} = p_\text{c}(S_\text{w})$, $k_{r\alpha}$ = $k_{r\alpha}(S_\text{w})$
* Physical constraint (void space filled with fluid phases): $S_\text{w} + S_\text{n} = 1$
* Primary variables: $p_\text{w}$, $S_\text{n}$ or $p_\text{n}$, $S_\text{w}$

* Further details can be found in the corresponding [documentation](https://dumux.org/docs/doxygen/master/group___two_p_model.html)

## 2pnc -- Two-Phase Compositional

* Transport equation for each component $\kappa \in \{\text{H2O}, \text{Air}, ...\}$ in phase $\alpha \in \{\text{w}, \text{n}\}$
$$
\begin{aligned}
\frac{\partial \left( \sum_\alpha \varrho_\alpha X_\alpha^\kappa \phi S_\alpha \right)}{\partial t} &+ \sum_\alpha \text{div}\left( \varrho_\alpha X_\alpha^\kappa \mathbf{v}_\alpha - \mathbf{D_{\alpha, pm}^\kappa} \varrho_\alpha \textbf{grad}\, X^\kappa_\alpha \right) = \sum_\alpha q_\alpha^\kappa 
\end{aligned}
$$

* Constitutive relation: $p_\text{c} := p_\text{n} - p_\text{w} = p_\text{c}(S_\text{w})$, $k_{r\alpha}$ = $k_{r\alpha}(S_\text{w})$
* Physical constraints: $S_\text{w} + S_\text{n} = 1$ and $\sum_\kappa X_\alpha^\kappa = 1$
* Primary variables: depending on the phase state

* Further details can be found in the corresponding [documentation](https://dumux.org/docs/doxygen/master/group___two_p_n_c_model.html)

## 2pncmin

* Transport equation for each component $\kappa \in \{\text{H2O}, \text{Air}, ...\}$
$$
\begin{aligned}
\frac{\partial \left( \sum_\alpha \varrho_\alpha X_\alpha^\kappa \phi S_\alpha \right)}{\partial t} &+ \sum_\alpha \text{div}\left( \varrho_\alpha X_\alpha^\kappa \mathbf{v}_\alpha - \mathbf{D_{\alpha, pm}^\kappa} \varrho_\alpha \textbf{grad}\, X^\kappa_\alpha \right)= \sum_\alpha q_\alpha^\kappa 
\end{aligned}
$$

* Mass balance solid phases
$$
\frac{\partial \left(\varrho_\lambda \phi_\lambda \right)}{\partial t} = q_\lambda \quad \forall \lambda \in \Lambda
$$
for a set of solid phases $\Lambda$ each with volume fraction $\varrho_\lambda$

* Source term models **dissolution/precipiation/phase transition** fluid &harr; solid

* Further details can be found in the corresponding [documentation](https://dumux.org/docs/doxygen/master/group___two_p_n_c_min_model.html)

## 3p -- Three-Phase Immiscible

* Uses standard multi-phase Darcy approach for the conservation of momentum by default
* Conservation of the phase mass of phase $\alpha \in \{\text{w}, \text{g}, \text{n}\}$
$$
\frac{\partial \left( \phi \varrho_\alpha S_\alpha \right)}{\partial t} - \text{div} \left( \varrho_\alpha \mathbf{v}_\alpha \right) = q_\alpha
$$

* Physical constraint: $S_\text{w} + S_\text{n} + S_g = 1$
* Primary variables: $p_\text{g}$, $S_\text{w}$, $S_\text{n}$

* Further details can be found in the corresponding [documentation](https://dumux.org/docs/doxygen/master/group___three_p_model.html)

## 3p3c -- Three-Phase Compositional

* Transport equation for each component $\kappa \in \{\text{H2O}, \text{Air}, \text{NAPL}\}$ in phase $\alpha \in \{\text{w}, \text{g}, \text{n}\}$
$$
\begin{aligned}
\frac{\partial \left( \phi \sum_\alpha \varrho_{\alpha,\text{mol}} x_\alpha^\kappa S_\alpha \right)}{\partial t}&+ \sum_\alpha \text{div} \left( \varrho_{\alpha,\text{mol}} x_\alpha^\kappa \mathbf{v}_\alpha - D_\text{pm}^\kappa \frac{1}{M_\kappa} \varrho_\alpha \textbf{grad} X^\kappa_{\alpha} \right) = q^\kappa 
\end{aligned}
$$

* Physical constraints: $\sum_\alpha S_\alpha = 1$ and $\sum_\kappa x^\kappa_\alpha = 1$
* Primary variables: depend on the locally present fluid phases

* Further details can be found in the corresponding [documentation](https://dumux.org/docs/doxygen/master/group___three_p_three_c_model.html)

## Other Porous-Medium Flow Models

* For other porous-medium flow models, we refer to the Doxygen documentation:

    - [2p1c](https://dumux.org/docs/doxygen/master/group___two_p_one_c_model.html)
    - [2p2c](https://dumux.org/docs/doxygen/master/group___two_p_two_c_model.html)
    - [3pwateroil](https://dumux.org/docs/doxygen/master/group___three_p_water_oil_model.html)
    - [co2](https://dumux.org/docs/doxygen/master/group___c_o2_model.html)
    - [mpnc](https://dumux.org/docs/doxygen/master/group___m_p_n_c_model.html)
    - [nonequilibrium](https://dumux.org/docs/doxygen/master/group___non_equilibrium_model.html)
    - [richards](https://dumux.org/docs/doxygen/master/group___richards_model.html)
    - [richardsnc](https://dumux.org/docs/doxygen/master/group___richards_n_c_model.html)
    - [tracer](https://dumux.org/docs/doxygen/master/group___tracer_model.html)

## Non-Isothermal (Equilibrium)

* Local thermal equilibrium assumption
* One energy conservation equation for the porous solid matrix and the fluids

    $$
    \begin{aligned}\frac{\partial \left( \phi \sum_\alpha \varrho_\alpha u_\alpha S_\alpha \right)}{\partial t} &+ \frac{\partial \left(\left(1 - \phi \right)\varrho_s c_s T \right)}{\partial t}
    + \sum_\alpha \text{div} \left( \varrho_\alpha h_\alpha \mathbf{v}_\alpha \right)
    - \text{div} \left(\lambda_\text{pm} \textbf{grad}\, T \right) = q^h \end{aligned}
    $$

* Specific internal energy $u_\alpha = h_\alpha - p_\alpha / \varrho_\alpha$
* Can be added to other models, additional primary variable temperature $T$

* Further details can be found in the corresponding [documentation](https://dumux.org/docs/doxygen/master/group___n_i_model.html)

## Free Flow (Navier-Stokes)

* Stokes equation
* Navier-Stokes equations
* Energy and component transport

## Reynolds-Averaged Navier-Stokes (RANS)

* Momentum balance equation for a single-phase, isothermal RANS model

    $$
    \frac{\partial \left(\varrho \textbf{v} \right)}{\partial t} + \nabla \cdot \left(\varrho \textbf{v} \textbf{v}^{\text{T}} \right) = \nabla \cdot \left(\mu_\textrm{eff} \left(\nabla \textbf{v} + \nabla \textbf{v}^{\text{T}} \right) \right)
    - \nabla p + \varrho \textbf{g} - \textbf{f}
    $$

* The effective viscosity is composed of the fluid and the eddy viscosity

    $$
    \mu_\textrm{eff} = \mu + \mu_\textrm{t}
    $$

* Various turbulence models are implemented

* More details can be found in the [Doxygen documentation](https://dumux.org/docs/doxygen/master/group___freeflow_models.html)

## Other Models

* For other models, we refer to the Doxygen documentation:

    - [Shallow water](https://dumux.org/docs/doxygen/master/group___shallow_water_models.html)
    - [Geomechanics](https://dumux.org/docs/doxygen/master/group___geomechanics_models.html)
    - [Pore network](https://dumux.org/docs/doxygen/master/group___pore_network_models.html)

## Your Model Equations?

# Spatial Discretization

## Cell-centered Finite Volume Methods

* Elements of the grid are used as control volumes
* Discrete **values** represent control volume average
* **Two-point flux approximation (TPFA)**
    * Simple and robust but not always consistent
* **Multi-point flux approximation (MPFA)**
    * A consistent discrete gradient is constructed

## Two-Point Flux Approximation (TPFA)

<img src="img/cctpfa.png" width="75%"/>

## Multi-Point Flux Approximation (MPFA)

<img src="img/ccmpfa.png" width="80%"/>

## Control-Volume Finite Element Methods

* Model domain is discretized using a **FE** mesh
* Secondary **FV** mesh is constructed &rarr; control volume/**box**
* Control volumes (CV) split into sub control volumes (SCVs)
* Faces of CV split into sub control volume faces (SCVFs)
* Unites advantages of finite-volume (simplicity) and finite-element methods (flexibility)
    * **Unstructured grids** (from FE method)
    * **Mass conservation** (from FV method)

## Box Method

Vertex-centered finite volumes / control volume finite element
method with piecewise linear polynomial functions ($\mathrm{P}_1/\mathrm{Q}_1$)

<img src="img/box.png" width="70%"/>

## Diamond Scheme 

Face-centered finite-volume scheme based on non-conforming finite-element spaces

<img src="img/fcdiamond.png" width="70%"/>

## PQ1 Bubble Scheme

Control-volume finite element scheme based on $\mathrm{P}_1/\mathrm{Q}_1$ basis functions with enrichment by a bubble function

<img src="img/pq1bubble.png" width="70%"/>

## Finite Volume Method on Staggered Control Volumes

* Uses a finite volume method with different staggered control volumes for different equations
* Fluxes are evaluated with a two-point flux approximation
* **Robust** and **mass conservative**
* Restricted to **structured grids** (tensor-product structure)

## Staggered Grid Discretization

<img src="img/fcstaggered.png"/>

# Model Components

## Model Components

* Typically, the following components have to be specified
    * **Model**: Equations and constitutive models
    * **Assembler**: Key properties (Discretization, Variables, LocalResidual)
    * **Solver**: Type of solution stategy (e.g. Newton)
    * **LinearSolver**: Method for solving linear equation systems (e.g. direct / Krylov subspace methods)
    * **Problem**: Initial and boundary conditions, source terms
    * **TimeLoop**: For time-dependent problems
    * **VtkOutputModule** / **IOFields**: For VTK output of the simulation

# Simulation Flow

## Simulation Flow

<img src="img/simulation_flow.png"/>