intro.md 3.61 KB
Newer Older
1
2
3
4
5
This tutorial was copied from dumux/test/porousmediumflow/tracer/1ptracer.

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

## Problem set-up
6
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.
7

8
![](./img/setup.png)
9

10
## Model description
11
12
Two different models are applied to simulate the system: In a first step, the groundwater velocity is evaluated under stationary conditions using the single phase model.
In a second step, the contaminant is transported with the groundwater velocity field. It is assumed, that the dissolved contaminant does not affect density and viscosity of the groundwater, and thus, it is handled as a tracer by the tracer model. The tracer model is then solved instationarily.
13
14

### 1p Model
15
The single phase model uses Darcy's law as the equation for the momentum conservation:
16

17
```math
18
\textbf v = - \frac{\textbf K}{\mu} \left(\textbf{grad}\, p - \varrho {\textbf g} \right),
19
```
20

21
with the darcy velocity $` \textbf v `$, the permeability $` \textbf K`$, the dynamic viscosity $` \mu`$, the pressure $`p`$, the density $`\rho`$ and the gravity $`\textbf g`$.
22

23
Darcy's law is inserted into the mass balance equation:
24

25
26
27
```math
\phi \frac{\partial \varrho}{\partial t} + \text{div} \textbf v = 0
```
28

29
where $`\phi`$ is the porosity.
30

31
The equation is discretized using a cell-centered finite volume scheme as spatial discretization for the pressure as primary variable. For details on the discretization scheme, have a look at the dumux [handbook](https://dumux.org/handbook).
32
33

### Tracer Model
34
The transport of the contaminant component $`\kappa`$ is based on the previously evaluated velocity field $`\textbf v`$  with the help of the following mass balance equation:
35

36
```math
37
\phi \frac{ \partial \varrho X^\kappa}{\partial t} - \text{div} \left\lbrace \varrho X^\kappa {\textbf v} + \varrho D^\kappa_\text{pm} \textbf{grad} X^\kappa \right\rbrace = 0,
38
```
39

40
where $`X^\kappa`$ is the mass fraction of the contaminant component $`\kappa`$ and $` D^\kappa_\text{pm} `$ is the effective diffusivity.
41

42
The effective diffusivity is a function of the diffusion coefficient of the component $`D^\kappa`$ and the porosity and tortuosity $`\tau`$ of the porous medium (see [dumux/material/fluidmatrixinteractions/diffusivityconstanttortuosity.hh](https://git.iws.uni-stuttgart.de/dumux-repositories/dumux/-/blob/master/dumux/material/fluidmatrixinteractions/diffusivityconstanttortuosity.hh)):
43

44
```math
45
D^\kappa_\text{pm}= \phi \tau D^\kappa.
46
```
47

48
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](https://dumux.org/handbook).
49

50
51
In the following, we take a close look at the files containing the set-up: The boundary conditions and spatially distributed parameters for the single phase model are set in `problem_1p.hh` and `spatialparams_1p.hh`.
For the tracer model, this is done in the files `problem_tracer.hh` and `spatialparams_tracer.hh`, respectively. Afterwards, we show the different steps for solving the model in the source file `main.cc`. Finally, some simulation results are shown.