@@ -3,7 +3,7 @@ This tutorial was copied from dumux/test/porousmediumflow/tracer/1ptracer.

# One-phase flow with random permeability distribution and a tracer model

## Problem set-up

This example contains a contaminant transported by a base groundwater flow in a randomly distributed permeability field. The figure below shows the simulation set-up. The permeability values range between 6.12e-15 and 1.5 e-7 $`m^2`$. A pressure gradient between the top an the bottom boundary leads to a groundwater flux from the bottom to the top. Neumann no-flow boundaries are assigned to the left and right boundary. Initially, there is a contaminant concentration at the bottom of the domain.

This example contains a contaminant transported by a base groundwater flow in a randomly distributed permeability field. The figure below shows the simulation set-up. The permeability values range between 6.12e-15 and 1.5 e-7 $`m^2`$. A pressure gradient between the top and the bottom boundary leads to a groundwater flux from the bottom to the top. Neumann no-flow boundaries are assigned to the left and right boundary. Initially, there is a contaminant concentration at the bottom of the domain.

![](./img/setup.png)

...

...

@@ -34,10 +34,10 @@ The equation is discretized using a cell-centered finite volume scheme as spatia

The transport of the contaminant component $`\kappa`$ is based on the previously evaluated velocity field $`\textbf v`$ with the help the following mass balance equation:

With the porosity $`\phi`$, the mass fraction of the contaminant component $`\kappa`$: $`X^\kappa`$, the porous medium diffusivity $` D^\kappa_\text{pm} `$, the molar masses of the component $` M^\kappa `$, the average molar mass of the phase $`M_\alpha`$ and the mole fraction $`x`$.

With the porosity $`\phi`$, the mass fraction of the contaminant component $`\kappa`$: $`X^\kappa`$, the porous medium diffusivity $` D^\kappa_\text{pm} `$ and the density of the fluid phase $`\varrho`$.

The porous medium diffusivity is a function of the diffusion coefficient of the component $`D^\kappa`$, the porosity $`\phi`$ and the porous medium tortuosity $`\tau`$ by the following equation:

...

...

@@ -45,7 +45,7 @@ The porous medium diffusivity is a function of the diffusion coefficient of the

D^\kappa_\text{pm}= \phi \tau D^\kappa

```

The primary variable of this model is the mass fraction $`X^\kappa`$. We apply the same spatial discretization as in the single pahse model and use the implicit Euler method for time discretization. For more information, have a look at the dumux handbook.

The primary variable of this model is the mass fraction $`X^\kappa`$. We apply the same spatial discretization as in the single phase model and use the implicit Euler method for time discretization. For more information, have a look at the dumux handbook.

In the following, we take a close look at the files containing the set-up: At first, boundary conditions and spatially distributed parameters are set in `problem_1p.hh` and `spatialparams_1p.hh`, respectively, for the single phase model and subsequently in `problem_tracer.hh` and `spatialparams_tracer.hh` for the tracer model. Afterwards, we show the different steps for solving the model in the source file `main.cc`. At the end, we show some simulation results.

@@ -3,9 +3,9 @@ This tutorial was copied from dumux/test/porousmediumflow/tracer/1ptracer.

# One-phase flow with random permeability distribution and a tracer model

## Problem set-up

This example contains a contaminant transported by a base groundwater flow in a randomly distributed permeability field. The figure below shows the simulation set-up. The permeability values range between 6.12e-15 and 1.5 e-7 $`m^2`$. A pressure gradient between the top an the bottom boundary leads to a groundwater flux from the bottom to the top. Neumann no-flow boundaries are assigned to the left and right boundary. Initially, there is a contaminant concentration at the bottom of the domain.

This example contains a contaminant transported by a base groundwater flow in a randomly distributed permeability field. The figure below shows the simulation set-up. The permeability values range between 6.12e-15 and 1.5 e-7 $`m^2`$. A pressure gradient between the top and the bottom boundary leads to a groundwater flux from the bottom to the top. Neumann no-flow boundaries are assigned to the left and right boundary. Initially, there is a contaminant concentration at the bottom of the domain.

![](./img/set-up.png)

![](./img/setup.png)

## Model description

Two different models are applied to simulate the system: In a first step, the groundwater velocity is evaluated under stationary conditions. Therefore the single phase model is applied.

...

...

@@ -34,10 +34,10 @@ The equation is discretized using a cell-centered finite volume scheme as spatia

The transport of the contaminant component $`\kappa`$ is based on the previously evaluated velocity field $`\textbf v`$ with the help the following mass balance equation:

With the porosity $`\phi`$, the mass fraction of the contaminant component $`\kappa`$: $`X^\kappa`$, the porous medium diffusivity $` D^\kappa_\text{pm} `$, the molar masses of the component $` M^\kappa `$, the average molar mass of the phase $`M_\alpha`$ and the mole fraction $`x`$.

With the porosity $`\phi`$, the mass fraction of the contaminant component $`\kappa`$: $`X^\kappa`$, the porous medium diffusivity $` D^\kappa_\text{pm} `$ and the density of the fluid phase $`\varrho`$.

The porous medium diffusivity is a function of the diffusion coefficient of the component $`D^\kappa`$, the porosity $`\phi`$ and the porous medium tortuosity $`\tau`$ by the following equation:

...

...

@@ -45,6 +45,6 @@ The porous medium diffusivity is a function of the diffusion coefficient of the

D^\kappa_\text{pm}= \phi \tau D^\kappa

```

The primary variable of this model is the mass fraction $`X^\kappa`$. We apply the same spatial discretization as in the single pahse model and use the implicit Euler method for time discretization. For more information, have a look at the dumux handbook.

The primary variable of this model is the mass fraction $`X^\kappa`$. We apply the same spatial discretization as in the single phase model and use the implicit Euler method for time discretization. For more information, have a look at the dumux handbook.

In the following, we take a close look at the files containing the set-up: At first, boundary conditions and spatially distributed parameters are set in `problem_1p.hh` and `spatialparams_1p.hh`, respectively, for the single phase model and subsequently in `problem_tracer.hh` and `spatialparams_tracer.hh` for the tracer model. Afterwards, we show the different steps for solving the model in the source file `main.cc`. At the end, we show some simulation results.