Commit a45fde2a by Katharina Heck

### [examples][1ptracer] fix tracer equation in readme to mass averaged

version of Fick's law and include density
parent 5eb4d4f6
 ... ... @@ -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: math \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 \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  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: math \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 \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  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.
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