intro.md 3.39 KB
 Gabriele Seitz committed Oct 07, 2019 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  Katharina Heck committed Oct 07, 2019 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.  Gabriele Seitz committed Oct 07, 2019 7   Katharina Heck committed Oct 07, 2019 8 ![](./img/setup.png)  Katharina Heck committed Oct 07, 2019 9   Gabriele Seitz committed Oct 07, 2019 10 ## Model description  Dennis Gläser committed Mar 19, 2020 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.  Gabriele Seitz committed Oct 07, 2019 13 14  ### 1p Model  Gabi Seitz committed Oct 07, 2019 15 The single phase model uses Darcy's law as the equation for the momentum conservation:  Gabriele Seitz committed Oct 07, 2019 16   Farid Mohammadi committed Oct 07, 2019 17 math  Dennis Gläser committed Mar 19, 2020 18 \textbf v = - \frac{\textbf K}{\mu} \left(\textbf{grad}\, p - \varrho {\textbf g} \right),  Farid Mohammadi committed Oct 07, 2019 19   Gabriele Seitz committed Oct 07, 2019 20   Dennis Gläser committed Mar 19, 2020 21 with the darcy velocity $ \textbf v $, the permeability $ \textbf K$, the dynamic viscosity $ \mu$, the pressure $p$, the density $\rho$ and the gravity $\textbf g$.  Gabi Seitz committed Oct 07, 2019 22   Dennis Gläser committed Mar 19, 2020 23 Darcy's law is inserted into the mass balance equation:  Gabriele Seitz committed Oct 07, 2019 24   Farid Mohammadi committed Oct 07, 2019 25 26 27 math \phi \frac{\partial \varrho}{\partial t} + \text{div} \textbf v = 0   Gabriele Seitz committed Oct 07, 2019 28   Dennis Gläser committed Mar 19, 2020 29 where $\phi$ is the porosity.  Gabi Seitz committed Oct 07, 2019 30   Farid Mohammadi committed Oct 07, 2019 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).  Gabriele Seitz committed Oct 07, 2019 32 33  ### Tracer Model  Dennis Gläser committed Mar 19, 2020 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:  Gabriele Seitz committed Oct 07, 2019 35   Farid Mohammadi committed Oct 07, 2019 36 math  Dennis Gläser committed Mar 19, 2020 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,  Farid Mohammadi committed Oct 07, 2019 38   Gabriele Seitz committed Oct 07, 2019 39   Dennis Gläser committed Mar 19, 2020 40 where $X^\kappa$ is the mass fraction of the contaminant component $\kappa$ and $ D^\kappa_\text{pm} $ is the effective diffusivity.  Gabriele Seitz committed Oct 07, 2019 41   Dennis Gläser committed Mar 19, 2020 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:  Gabriele Seitz committed Oct 07, 2019 43   Farid Mohammadi committed Oct 07, 2019 44 math  Dennis Gläser committed Mar 19, 2020 45 D^\kappa_\text{pm}= \phi \tau D^\kappa.  Farid Mohammadi committed Oct 07, 2019 46   Gabi Seitz committed Oct 07, 2019 47   Dennis Gläser committed Mar 19, 2020 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).  Gabi Seitz committed Oct 07, 2019 49   Dennis Gläser committed Mar 19, 2020 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.