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intro.md 11.81 KiB 
title: Introduction to DuMu^x^
subtitle: Overview and Available Models

Table of Contents

Table of Contents

  1. History and Structure
  2. Available Models
  3. Spatial Discretization
  4. Model Components
  5. Simulation Flow

History and Structure

The DuMu^x^ Development Team (IWS-LH2)

DuMu^x^ is a DUNE module

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

  • DuMu^x^: DUNE for Multi-{Phase, Component, Scale, Physics, \text{...}} flow and transport in porous media.
  • Goal: sustainable and consistent framework for the implementation and application of model concepts and constitutive relations.
  • Meanwhile developed by more than 30 PhD students and post docs, mostly at LH2.

Application

  • Successfully applied to
    • gas (CO2, H2, CH4, ...) storage scenarios
    • environmental remediation problems
    • transport of therapeutic agents through biological tissue
    • root-soil interaction
    • subsurface-atmosphere coupling (Navier-Stokes / Darcy)
    • pore-network modelling
    • flow and transport in fractured porous media

Exemplary DuMu^x^ Modules

  • dumux-lecture: example applications for lectures offered by LH2 in Stuttgart
  • dumux-pub: accompany a publication with all code and data to reproduce the results
  • dumux-appl: current unpublished development and ongoing research

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, ..., 10/2017: Release 2.12.
  • 09/2015: Transition from Subversion to Git.
  • 12/2018: Release 3.0, ..., 03/2023: Release 3.7.

Downloads and Publications

  • More than 1000 "real" and unique release downloads.
  • More than 200 peer-reviewed publications and PhD theses.

Evolution of C++ Files

Evolution of Code Lines

Mathematical Models

  • Porous medium flow (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 equation.
  • Shallow water flow: Two-dimensional shallow water flow (depth-averaged).
  • Geomechanics: Models taking into account solid deformation.
  • Pore network: Single and multi-phase models for flow and transport in pore networks.

Mailing Lists and GitLab

Documentation

Available Models

Available Models

Darcy's law

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

  • One-phase flow

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

  • Multi-phase flow (phase \alpha)

    v_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left(\textbf{grad}\, p_\alpha - \varrho_\alpha \mathbf{g} \right)

1p -- single-phase

  • Uses standard Darcy approach for the conservation of momentum

  • Mass continuity equation

    \phi \frac{\partial \varrho}{\partial t} + \text{div} \left\lbrace - \varrho \frac{\textbf{K}}{\mu} \left(\textbf{grad}\, p - \varrho \textbf{g} \right) \right\rbrace = q

  • Primary variable: p

1pnc -- single-phase, multi-component

  • Uses standard Darcy approach for the conservation of momentum

  • Transport of component \kappa \in \{w, a, ...\}

    \phi \frac{\partial \varrho X^\kappa}{\partial t} - \text{div} \left\lbrace \varrho X^\kappa \frac{\textbf {K}}{\mu} \left(\textbf{grad}\, p - \varrho \textbf{g} \right) + \varrho D^\kappa_\text{pm} \textbf{grad} X^\kappa \right\rbrace = q

  • Primary variables: p and x^\kappa

1pncmin -- with mineralization

  • Transport equation for each component \kappa \in \{w, a, ...\}

    \frac{\partial \left( \varrho_f X^\kappa \phi \right)}{\partial t} - \text{div} \left\lbrace \varrho_f X^\kappa \frac{k_{r}}{\mu} \mathbf{K} \left(\textbf{grad}\, p - \varrho_f \mathbf{g} \right) \right\rbrace - \text{div} \left\lbrace \mathbf{D_{pm}^\kappa} \varrho_f \textbf{grad}\, X^\kappa \right\rbrace = q_\kappa

  • Mass balance solid or mineral phases

    \frac{\partial \left(\varrho_\lambda \phi_\lambda \right)}{\partial t} = q_\lambda

  • Primary variables: p, x^k and \phi_\lambda

2p -- two-phase

  • Uses standard multi-phase Darcy approach for the conservation of momentum

  • Conservation of the phase mass of phase \alpha \in \{w, n\}

    \phi \frac{\partial \varrho_\alpha S_\alpha}{\partial t} - \text{div} \left\{\varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left(\textbf{grad}\, p_\alpha - \varrho_\alpha \mathbf{g} \right) \right\} = q_\alpha

  • Constitutive relation p_c = p_n - p_w

  • S_w + S_n = 1

  • Primary variables: p_w and S_n or p_n and S_w

2pnc

  • Transport equation for each component \kappa \in \{w, n, ...\} in phase \alpha \in \{w, n\}

    \frac{\partial \sum_\alpha \varrho_\alpha X_\alpha^\kappa \phi S_\alpha}{\partial t} - \sum_\alpha \text{div} \left\lbrace \varrho_\alpha X_\alpha^\kappa \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left( \textbf{grad}\, p_\alpha - \varrho_\alpha \mathbf{g} \right) \right\rbrace - \sum_\alpha \text{div} \left\lbrace \mathbf{D_{\alpha, pm}^\kappa} \varrho_\alpha \textbf{grad}\, X^\kappa_\alpha \right\rbrace = \sum_\alpha q_\alpha^\kappa

  • Constitutive relation p_c = p_n - p_w

  • S_w + S_n = 1 and X^\kappa_w + X^\kappa_n = 1

  • Primary variables: depend on the phase state

2pncmin

  • Transport equation for each component \kappa \in \{w, n, ...\} in phase \alpha \in \{w, n\}

    \frac{\partial \sum_\alpha \varrho_\alpha X_\alpha^\kappa \phi S_\alpha}{\partial t} - \sum_\alpha \text{div} \left\lbrace \varrho_\alpha X_\alpha^\kappa \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left( \textbf{grad}\, p_\alpha - \varrho_\alpha \mathbf{g} \right) \right\rbrace - \sum_\alpha \text{div} \left\lbrace \mathbf{D_{\alpha, pm}^\kappa} \varrho_\alpha \textbf{grad}\, X^\kappa_\alpha \right\rbrace = \sum_\alpha q_\alpha^\kappa

  • Mass balance solid or mineral phases

    \frac{\partial \left(\varrho_\lambda \phi_\lambda \right)}{\partial t} = q_\lambda

  • p_c = p_n - p_w, S_w + S_n = 1 and X^\kappa_w + X^\kappa_n = 1

  • Primary variables: depend on the phase state

3p -- three-phase

  • Uses standard multi-phase Darcy approach for the conservation of momentum

  • Conservation of the phase mass of phase \alpha \in \{w, g, n\}

    \phi \frac{\partial \varrho_\alpha S_\alpha}{\partial t} - \text{div} \left\lbrace \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left(\textbf{grad}\, p_\alpha - \varrho_\alpha \mathbf{g} \right) \right\rbrace = q_\alpha

  • S_w + S_n + S_g = 1

  • Primary variables: p_g, S_w and S_n

3p3c

  • Transport equation for each component \kappa \in \{w, a, c\} in phase \alpha \in \{w, g, n\}

    \phi \frac{\partial \left(\sum_\alpha \varrho_{\alpha,mol} x_\alpha^\kappa S_\alpha \right)}{\partial t} - \sum_\alpha \text{div} \left\lbrace \frac{k_{r\alpha}}{\mu_\alpha} \varrho_{\alpha,mol} x_\alpha^\kappa \mathbf{K} \left(\textbf{grad}\, p_\alpha - \varrho_{\alpha,mass} \mathbf{g} \right) \right\rbrace - \sum_\alpha \text{div} \left\lbrace D_\text{pm}^\kappa \frac{1}{M_\kappa} \varrho_\alpha \textbf{grad} X^\kappa_{\alpha} \right\rbrace = q^\kappa

  • S_w + S_n + S_g = 1 and x^w_\alpha + x^a_\alpha + x^c_\alpha = 1

  • Primary variables: depend on the locally present fluid phases

Non-Isothermal

  • Local thermal equilibrium is assumed

  • One energy conservation equation for the porous solid matrix and the fluids

    \phi \frac{\partial \sum_\alpha \varrho_\alpha u_\alpha S_\alpha}{\partial t} + \left(1 - \phi \right) \frac{\partial \left(\varrho_s c_s T \right)}{\partial t} - \sum_\alpha \text{div} \left\lbrace \varrho_\alpha h_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left(\textbf{grad}\, p_\alpha - \varrho_\alpha \mathbf{g} \right) \right\rbrace - \text{div} \left(\lambda_{pm} \textbf{grad}\, T \right) = q^h

  • u_\alpha = h_\alpha - p_\alpha / \varrho_\alpha

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}

Spatial Discretization

Cell Centered Finite Volume Methods

  • Use elements of the grid as control volumes
  • Discrete values are determined at the element/control volume center
  • Two-point flux approximation (TPFA)
    • Simple but robust
  • Multi-point flux approximation (MPFA)
    • A consistent discrete gradient is constructed

Two-Point Flux Approximation (TPFA)

Multi-Point Flux Approximation (MPFA)

Box method

  • Model domain is discretized using a FE mesh
  • Secondary FV mesh is constructed → control volume/box
  • Control volumes are partitioned into sub-control volumes (scvs)
  • Faces of control volumes are partitioned into sub-control volume faces (scvfs)
  • Unites advantages of finite-volume and finite-element methods
    • Unstructured grids (from FE method)
    • Mass conservative (from FV method)

Box method

Staggered Grid

  • Uses a finite volume method with different control volumes for different equations
  • Fluxes are evaluated with a two-point flux approximation
  • Robust and mass conservative
  • Should be applied for structured grids only

Staggered Grid

Model Components

Model Components

  • The following components have to be specified
    • Solver: Type of solution stategy
    • Assembler: Key properties
      • Geometry, Variables, LocalResidual
    • LinearSolver: How to solve algebraic equations
    • Problem: Initial and boundary conditions
    • SolutionVector: Container to store the solution
    • TimeLoop: For time-dependent problems
    • IOFields and VtkOutputModule: Output of the simulation

Simulation Flow

Simulation Flow