Commit dff511b7 authored by Gabi Seitz's avatar Gabi Seitz Committed by Katharina Heck
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[examples][1ptracer] add some transitional phrases to better guide through the...

[examples][1ptracer] add some transitional phrases to better guide through the README and revise intro.md
parent 08ddfdb5
......@@ -3,37 +3,42 @@ 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 2d simulation of a stationary groundwater flow. The permeability is distributed randomly. An contaminant initially concentrated at the domain bottom gets transported by the base groundwater flow.
The image below shows the simulation set-up. 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 domain bottom.
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 domain bottom.
<img src="Plots/setup.png" width="500">
## Model description
Two different models are applied to simulate the system: In the first step, the groundwater velocity is evaluated under stationary conditions. Therefore the single phase model is applied. In a second step, the contaminant gets transported based on the volume fluxes of the single phase flow. It is assumed, that the dissolved contaminant does not affect density and viscosity of the groundwater. The tracer model is solved instationarily.
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.
In a second step, the contaminant gets transported based on the on 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.
### 1p Model
The single phase model uses Darcy's law as the equation for the conservation of momentum:
The single phase model uses Darcy's law as the equation for the momentum conservation:
$` \textbf v = - \frac{\textbf K}{\mu} \left(\textbf{grad}\, p - \varrho {\textbf g} \right) `$
With the darcy velocity $` \textbf v `$, the permeability $` \textbf K`$, the viscosity $` \mu`$, the pressure $`p`$, the density $`\rho`$ and the gravity $`\textbf g`$.
Darcy's law is inserted into the continuity equation:
$` \phi \frac{\partial \varrho}{\partial t} + \text{div} \textbf v = 0`$
with density $`\rho`$. It is solved for the pressure as primary variable. This equation is discretized using a cell-centered finite volume scheme as spatial discretization. For details on the discretization, we refer to the dumux handbook.
with porosity $`\phi`$.
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.
### Tracer Model
With the velocity field $`\textbf v`$ the transport of the contaminant component $`\kappa`$ is described by the following equation:
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:
$` \phi \frac{ \partial X^\kappa}{\partial t} - \text{div} \left\lbrace X^\kappa {\textbf v}+ D^\kappa_\text{pm} \frac{M^\kappa}{M_\alpha} \textbf{grad} x^\kappa \right\rbrace = 0 `$
With the porosity $`\phi`$, the mass fraction of the component $`\kappa`$: $`X^\kappa`$, the binary diffusion coefficient in the porous medium $` D^\kappa_\text{pm} `$, the molar masses $` M `$ of the component $`\kappa`$ and the phase $`\alpha`$ and the mole fraction $`x`$.
The primary variable of this model is the mass fraction $`X^\kappa`$.
With the porosity $`\phi`$, the mass fraction of the contaminant component $`\kappa`$: $`X^\kappa`$, the binary diffusion coefficient in the porous medium $` D^\kappa_\text{pm} `$, the molar masses of the component $` M^\kappa `$, the average molar mass of the phase $`M_\alpha`$ and the mole fraction $`x`$.
The porous medium diffusivity is yield out of the diffusion coefficient of the component, the porosity $`\phi `$ and the porous medium tortuosity $`\tau`$ by the following equation:
$`
D^\kappa_\text{pm}= \phi \tau D^\kappa
`$
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.
In the following, we take a close look at the files containing the setup: 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.
\ No newline at end of file
......@@ -16,10 +16,14 @@
* You should have received a copy of the GNU General Public License *
* along with this program. If not, see <http://www.gnu.org/licenses/>. *
*****************************************************************************/
// ## Header guard
#ifndef DUMUX_ONEP_TRACER_TEST_PROBLEM_HH
#define DUMUX_ONEP_TRACER_TEST_PROBLEM_HH
//Before we enter the problem class containing initial and boundary conditions, we include necessary files and introduce properties.
// ## Include files
// The dune grid interphase is included here:
......@@ -176,5 +180,5 @@ public:
};
// We leave the namespace Dumux.
}
} // end namespace Dumux
#endif
......@@ -21,6 +21,8 @@
#ifndef DUMUX_TRACER_TEST_PROBLEM_HH
#define DUMUX_TRACER_TEST_PROBLEM_HH
//Before we enter the problem class containing initial and boundary conditions, we include necessary files and introduce properties.
// ## Include files
// Again, we have to include the dune grid interphase:
......@@ -208,6 +210,6 @@ private:
};
// We leave the namespace Dumux here.
}
} // end namespace Dumux
#endif
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