Commit 616569ed by Johannes Hommel Committed by Katharina Heck

### [EFM2] updated exercise2

parent c62db710
 ... ... @@ -7,7 +7,7 @@ Diffusive forces are to be neglected in this part. \item What are the driving forces that should be considered? \item Guess how long the different parts of the process will take. \item Which equation(s) do you need to describe the problem? Which assumptions do you make? \item How much DNAPL would be necessary, so that it enters the groundwater system? Is that a realistic amount? Use the parameters given in section \ref{prop_paramI}. \item How much DNAPL would be necessary, so that it enters the groundwater system? Is that a realistic amount given the leaky car engine setup? Use the parameters given in section \ref{prop_paramI}. \end{enumerate} \section{Numerical Simulation} ... ... @@ -20,7 +20,10 @@ Diffusive forces are to be neglected in this part. \end{enumerate} \subsection{Exercise} Simulate the given problem. Simulate the given problem.\\ Open a terminal and type {\em cd /temp/efm2019/ex2}\\ Run the simulation: {\em ./lens2pexercise}\\ \begin{enumerate} \item Find out how long the different parts of the process will take. \item Try different soil properties and see how they affect the simulation. ... ... @@ -28,7 +31,7 @@ Simulate the given problem. \end{enumerate} \clearpage \subsection{Properties and parameters - exercise3.input} \subsection{Properties and parameters - lens2pexercise2.input} \label{prop_paramI} {\scriptsize \lstset{numbers=left, breaklines=true, morecomment=[l]{\#}, commentstyle=\color{blue}\tiny, breakindent=28em} ... ... @@ -39,8 +42,8 @@ Simulate the given problem. \begin{tabular}[t]{llll} $\rho_n=$ & non-wetting phase density & $1.46 \cdot 10^{3}$ & [kg/m$^3$] \\ $\rho_w=$ & wetting phase density & $1.0 \cdot 10^{3}$ & [kg/m$^3$] \\ %$\Phi_1=$ & porosity in area one & $0.4$ & [-] \\ %$\Phi_2=$ & porosity in area two & $0.38$ & [-] \\ $\Phi_1=$ & porosity in area one & $0.4$ & [-] \\ $\Phi_2=$ & porosity in area two & $0.38$ & [-] \\ %$\mathbf{K}_1=$ & permeability in area one& $1.0 \cdot 10^{-10} \cdot \mathbf{I}$ & [m$^2$] \\ %$\mathbf{K}_2=$ & permeability in area two & $1.0 \cdot 10^{-13} \cdot \mathbf{I}$ & [m$^2$] \\ $\mathrm{g}=$ & gravitational acceleration & $9.806$ & [m/s$^2$] \\ ... ...
 ... ... @@ -35,17 +35,17 @@ p_D=p_n -p_w = h \mathrm{g} (\rho_n-\rho_w) \label{volume} V = 3 \cdot 3\cdot h \qquad \mathrm{m}^3 V_\mathrm{1D} = \Phi_1 \cdot 3 \cdot 3\cdot h \qquad \mathrm{m}^3 V_\mathrm{half-sphere} = \Phi_1 \cdot \frac{2}{3} \cdot \pi \cdot h^3 \qquad \mathrm{m}^3 {\em It is assumed, that the DNAPL cannot infiltrate the less permeable layer by diffusion, but nevertheless it is distributed in the domain equally, i.e. (\ref{volume}) holds at any time. The equations do not account for saturation, i.e. it is assumed that the DNAPL pool only consists of DNAPL ($S_n=1.0$).} \item How much DNAPL would be necessary, so that it enters the groundwater system? Is that a realistic amount? \\ {\em Equation (\ref{pd}) gives a height of 0.277m to overcome the entry pressure of 1250 Pa. That leads to an amount of 2494$l$ oil in the 3m by 3m domain. If a large limousine with a total engine oil of ca. 8$l$ is assumed, that is not realistic. Assuming still a constant distribution in the x-y--plane and a radial spreading, the plume must not spread more than 0.096m ($r=\sqrt{V \, / (\pi h)}$) to generate a pressure large enough to make the DNAPL enter the less permeable layer.} \item How much DNAPL would be necessary, so that it enters the groundwater system? Is that a realistic amount given the leaky car engine setup? \\ {\em Equation (\ref{pd}) gives a height of 0.11m to overcome the entry pressure of 500 Pa. with a 1D assumption, that leads to an amount of 399l oil in the 3m by 3m domain. However, on the other extreme, assuming that the oil spreads radially in every direction in the form of a half sphere, the maximum volume is much smaller. In this case, only 1.1l are needed to reach the entry pressure.} \end{enumerate} \clearpage ... ... @@ -111,7 +111,7 @@ $q_w$ & mass source/sink rate for non-wetting phase & [kg/(m$^3$s)] \item What simplifications can you you use to get a reduction of computational time?\\ {\em Reduce the dimension of the domain to the necessary size, i.e. adapt the length/width/height, choose a 2D domain or a radial symmetric cake'' domain.} {\em Reduce the dimension of the domain to the necessary size, i.e. adapt the length/width/height, choose a 1D or 2D domain or a radial symmetric cake'' domain.} \item Which boundary conditions would you use?\\ ... ... @@ -138,6 +138,6 @@ It is not at all easy to guess in an analytical approach the result for a more c %\end{enumerate} \begin{figure}[ht]\centering \includegraphics[width=0.7\linewidth]{pics/result3.png} \caption{$S_{\mathrm{n}}$ at 19145s with the parameters given in \ref{prop_paramI}.} \includegraphics[width=0.7\linewidth]{pics/resultWLense.png} \caption{$S_{\mathrm{n}}$ at 10000s with the parameters given in \ref{prop_paramI}.} \end{figure} \ No newline at end of file