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 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  Felix Weinhardt 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 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.  Gabriele Seitz committed Oct 07, 2019 7   Theresa Schollenberger committed Oct 07, 2019 8   Gabriele Seitz committed Oct 07, 2019 9 10  ## Model description  Gabi Seitz committed Oct 07, 2019 11 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.  Felix Weinhardt committed Oct 07, 2019 12 In a second step, the contaminant gets transported based 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.  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   Gabi Seitz committed Oct 07, 2019 17 $ \textbf v = - \frac{\textbf K}{\mu} \left(\textbf{grad}\, p - \varrho {\textbf g} \right) $  Gabriele Seitz committed Oct 07, 2019 18   Felix Weinhardt committed Oct 07, 2019 19 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 20   Gabriele Seitz committed Oct 07, 2019 21 22 Darcy's law is inserted into the continuity equation:  Gabi Seitz committed Oct 07, 2019 23 $ \phi \frac{\partial \varrho}{\partial t} + \text{div} \textbf v = 0$  Gabriele Seitz committed Oct 07, 2019 24   Gabi Seitz committed Oct 07, 2019 25 26 27 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.  Gabriele Seitz committed Oct 07, 2019 28 29  ### Tracer Model  Gabi Seitz committed Oct 07, 2019 30 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:  Gabriele Seitz committed Oct 07, 2019 31   Gabi Seitz committed Oct 07, 2019 32 $ \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 $  Gabriele Seitz committed Oct 07, 2019 33   Gabi Seitz committed Oct 07, 2019 34 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 $x$.  Gabriele Seitz committed Oct 07, 2019 35   Gabi Seitz committed Oct 07, 2019 36 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:  Gabriele Seitz committed Oct 07, 2019 37   Bernd Flemisch committed Oct 07, 2019 38 $  Gabi Seitz committed Oct 07, 2019 39 D^\kappa_\text{pm}= \phi \tau D^\kappa  Bernd Flemisch committed Oct 07, 2019 40 $  Gabi Seitz committed Oct 07, 2019 41 42 43  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.  Theresa Schollenberger committed Oct 07, 2019 44 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.