In this example, we create a diffusion equation model and then simulate a diffusive process.

You learn how to

• setup a new simple model equation (diffusion equation)
• read parameters from a configuration file
• create a type tag and specialize properties for it
• generate a randomly distributed initial field (with MPI parallelism)
• solve a time-dependent diffusion problem in parallel

Model equations: A diffusion equation model fully developed and contained within the example
Discretization method: Vertex-centered finite volumes / control-volume finite elements (Lagrange, P1) (BoxModel)

In this example we simulate the separation of two immiscible phases using the Cahn-Hilliard equation.

You learn how to

• setup a new nonlinear model equation (Cahn-Hilliard equation)
• setup a model with two governing equations
• implement a custom volume variables class to name variables
• generate a randomly distributed vector-valued initial field (with MPI parallelism)
• solve a time-dependent diffusion problem in parallel

Model equations: A Cahn-Hilliard equation model fully developed and contained within the example
Discretization method: Vertex-centered finite volumes / control-volume finite elements (Lagrange, P1) (BoxModel)

In this example, we simulate tracer transport through a confined aquifer with a randomly distributed permeability field. We first solve the pressure field, compute the steady state flow field, and then solve the tracer transport equation.

You learn how to

• generate a randomly distributed permeability field
• sequentially solve two types of problems after each other:
  • solve a one-phase flow in porous media problem
  • compute the flow field from a pressure solution to pass to a tracer problem
  • solve an unsteady tracer transport problem with a given flow field

Model equations: Single-phase flow Darcy equation and advection-diffusion equation in porous media (OneP and Tracer)
Discretization method: Cell-centered finite volumes with two-point flux approximation (CCTpfaModel)

In this example we model a soil contamination problem where a DNAPL (dense non-aqueous phase liquid) infiltrates a water-saturated porous medium (two-phase flow). The initial distribution of DNAPL is read in from a txt-file. The grid is adaptively refined where DNAPL enters the domain, around the plume, and around an injection well.

You learn how to

• solve a two-phase flow in porous media problem with two immiscible phases
• set boundary conditions and a simple injection well
• implement a problem with heterogeneous material parameters
• use adaptive grid refinement around the saturation front

Model equations: Immiscible two-phase flow Darcy equations in porous media (TwoP)
Discretization method: Cell-centered finite volumes with two-point flux approximation (CCTpfaModel)

The shallow water flow model is applied to simulate steady subcritical flow in a channel including a bottom friction model.

You learn how to

• solve a shallow water flow problem including bottom friction
• compute and output (VTK) an analytical reference solution

Model equations: 2D shallow water equations (ShallowWater)
Discretization method: Cell-centered finite volumes with Riemann solver (CCTpfaModel)

In this example, we simulate a free flow between two plates in two dimensions.

You learn how to

• solve a free flow problem
• set outflow boundary conditions in the free-flow context

Model equations: 2D Stokes equations (NavierStokes)
Discretization method: Finite volumes with face-centered degrees of freedom of the staggered grid arrangement (FaceCenteredStaggeredModel)

In this example, a rotation-symmetric solution for the single-phase flow equation is discussed.

You learn how to

• solve a rotation-symmetric problem
• perform a convergence test against an analytical solution
• do post-processing in ParaView

Model equations: (rotation-symmetric) single-phase flow Darcy equation (OneP)
Discretization method: Vertex-centered finite volumes / control-volume finite elements (Lagrange, P1) (BoxModel)

In this example, we simulate microbially-induced calcite precipitation.

You learn how to

• solve a reactive transport model including
  • biofilm growth
  • mineral precipitation and dissolution
  • changing porosity and permeability
• use complex fluid and solid systems
• set a complex time loop with checkpoints, reading the check points from a file
• set complex injection boundary conditions, reading the injection types from a file

Model equations: Miscible two-phase multi-component flow Darcy equations with precipitation and reaction (TwoPNCMin)
Discretization method: Vertex-centered finite volumes / control-volume finite elements (Lagrange, P1) (BoxModel)

In this example, we simulate laminar incompressible flow in a cavity with the Navier-Stokes equations.

You learn how to

• solve a single-phase Navier-Stokes flow problem
• compare the results of Stokes flow (Re = 1) and Navier-Stokes flow (Re = 1000)
• compare the numerical results with the reference data using the plotting library matplotlib

Model equations: Navier-Stokes equations (NavierStokes)
Discretization method: Finite volumes with staggered grid arrangement (StaggeredFreeFlowModel)

In this example, we use a single-phase flow pore-network model to estimate the upscaled Darcy permeability of a randomly generated grid.

You learn how to

• solve a single-phase-flow pore-network problem
• use the total mass flow rate to estimate \(K_{xx}\), \(K_{yy}\), \(K_{zz}\)

Model equations: Single-phase flow pore-network model (PNMOneP)
Discretization method: Pore-network (PoreNetworkModel)

In this example, we compute the spread of a tracer in the blood stream and the embedding tissue (porous medium). We couple a 1D advection-diffusion equation on the network with a 3D diffusion equation in the embedding porous medium.

You learn how to

• setup a multi-domain with domains of different dimension
• read data from a grid file
• specify input parameters in multi-domain simulations

Model equations: 1D and 3D advection-diffusion equations (Tracer)
Discretization method: Cell-centered finite volumes with two-point flux approximation (CCTpfaModel)