\section{Meyer-Peter-Muller transport formula for total transport \label{case:mpm01}}
\newrefsegment

%---------------------------------------------------------------------------------
\section*{Quality Assurance}
{{\footnotesize
\begin{tabular}{p{22mm-2pt}@{}p{27mm-2pt}@{}p{18mm}@{}p{27mm-2pt}@{}p{18mm}@{}p{27mm-2pt}@{}p{\textwidth-139mm-1pt}} \hline
    \rowcolor{dgrey4}
\textbf{Date} & \textbf{Author} & \textbf{Initials} & \textbf{Review} & \textbf{Initials} & \textbf{Approval} & \textbf{Initials}  \\ \hline
01 June 2018 & Andries Paarlberg & & Willem Ottevanger & & Aukje Spruyt &   \\ \hline
  \end{tabular}
}}

%---------------------------------------------------------------------------------
\section*{Version information}
\begin{tabular}{@{}p{23mm}@{}p{0mm}p{\textwidth-23mm-30pt}}
\textbf{Date of study}& \textbf{:} & 01 June 2018 \\
\textbf{Executable}   & \textbf{:} & D-Flow FM Version 1.2.0.55920M, May 31 2018, 20:05:27 \\
\textbf{Location}     & \textbf{:} & \url{https://svn.oss.deltares.nl/repos/trunk/riverlab/f27_mor1d_tabulated_crosssections/c02_mc_sediment_transport_MPM} \\ 
\textbf{SVN revision} & \textbf{:} & -
\end{tabular}

%---------------------------------------------------------------------------------
\section*{Purpose}
\DFLOWFM-1D-mor can use so-called tabulated cross-sections (from SOBEK3), where the flow width is specified as a function of the water level. The bed level is updated in the main channel only. \\
The purpose of this validation case is to prove that the calculated sediment transport with the \citet{MeyerPeterMuller48} formula is correct in \DFLOWFM-1D, with tabulated cross sections. The formula is tested using several different parameter values, as outlined below.

%---------------------------------------------------------------------------------
\section*{Linked claims}
\begin{itemize}
\item{Water levels computed with \DFLOWFM are comparable to water levels computed with SOBEK3.}
\item{\DFLOWFM accurately calculates sediment transport according to the formula of Meyer-Peter-Muller.}
\end{itemize}

\section*{Approach}
We start from an existing SOBEK3-model (\url{/DSCTestbench/cases/e106_dflow1d/f13_morphology/c02_sediment_transport_Meyer-Peter-Muller}). 
%The values of the Ch\'ezy roughness, the bed slope $i_b$ and the sediment grain size are varied. 
The values of the Ch\'ezy roughness, and the bed slope $i_b$ are varied. 
The test is carried out for several different discharge values (and downstream water levels). In this test case, we assume that the velocities as calculated by the flow module are correct; simulated water levels are compared with SOBEK3.

The total sediment transport $S $ \unitbrackets{kg/s} calculated by the Engelund-Hansen formula is given by:
\begin{align}\label{eq:mpm}
	S = B \rho_s s = B \rho_s (0.05\frac{\alpha u^5}{\sqrt{g}C^3\Delta^2D_{50}})
\end{align}
where 
\begin{symbollist}
\item[$u$]  the magnitude of the flow velocity \unitbrackets{m/s}, 
\item[$\Delta$]  the relative density given by $\Delta = \frac{\rho_s-\rho_w}{\rho_w}$ \unitbrackets{-}, 
\item[$C$]  the Ch\'ezy friction coefficient \unitbrackets{m$^{1/2}$/s}, 
\item[$D_{50}$] the median diameter of sediment \unitbrackets{m}, 
\item[$\alpha$] the user-defined calibration coefficient \unitbrackets{-}, 
\item[$B$] the width of the main channel \unitbrackets{m} and 
\item[$\rho_s$] the sediment density \unitbrackets{kg/m$^3$}. 
\end{symbollist}
%
In this test, the sediment transport is calculated analytically based on the calculated velocity by the flow module. The analytically computed sediment transport is then compared to the sediment transport as calculated by \DFLOWFM-1D.

%---------------------------------------------------------------------------------
\section*{Model description}
The model domain, consisting of four (non-connected) straight branches, each with the same length of 10 km, but different properties is shown in \autoref{fig:mpm_4_branches}. 
%
\begin{figure}[H]
	\includegraphics[width=\textwidth]{figures/4takken.png}
	\caption{Four (non-connected) branches, channels with equal length, but different properties.}
	\label{fig:mpm_4_branches}
\end{figure}
%
For this test case, each of the channels is straight with homogeneous cross-sections. The flow is stationary but non-homogeneous. 
The flow direction in the channel is oriented in the same direction as the computational grid. Upstream a discharge boundary is imposed (\autoref{fig:mpm_upstream_discharge}), downstream a water level boundary (\autoref{fig:mpm_downstream_water_level}). % add description or plot of boundary conditions
No further connection nodes are added. The channel contains rectangular cross-sections of 200 m in width throughout. See \autoref{fig:mpm_profile}. Computational nodes are placed at non-uniform distances. Calculations are done for four different testcases in which several parameter values are varied. An overview of these varied parameter values is shown in \autoref{tab:mpm_parametervalues}. Note that T1 and T3, two independent branches, have identical settings and should produce identical results. Other model settings for all cases are given in \autoref{tab:modeltestcase1}.
%
\begin{figure}[H]
	\includegraphics[width=\textwidth]{figures/Q_t_boundary_condition.jpg}
	\caption{Upstream discharge boundary condition.}
	\label{fig:mpm_upstream_discharge}
\end{figure}
%
\begin{figure}[H]
	\includegraphics[width=\textwidth]{figures/h_t_boundary_condition.jpg}
	\caption{Downstream water level boundary condition.}
	\label{fig:mpm_downstream_water_level}
\end{figure}
%
\begin{figure}[H]
	\includegraphics[width=\textwidth]{figures/rectangular_profile.jpg}
	\caption{Cross-section of test case.}
	\label{fig:mpm_profile}
\end{figure}

%
The basis model we use is set-up in SOBEK3. The geometry (tabulated cross-sections, roughness and some other properties), are directly imported into \DFLOWFM using a specific keyword in the MDU-file:
\begin{verbatim} OneDNetworkFile = dflow1d\water flow 1d.md1d \end{verbatim}
Some settings have to be set manually though:
\begin{itemize}
\item{A SOBEK3 import only works if a (dummy) Network file is present. We advise to use a 2D network, to ensure that it doesn't conflict with the 1D network (as imported from SOBEK3).}
\item{Timings such as refdate, start/end time and map-output intervals need to be converted manually to the MDU-file.}
\item{The boundary conditions (locations and values) are not yet imported from SOBEK3. They have to be manually created. We defined 4 polylines at the upstream boundary for imposing the discharge, and 4 polylines at the downstream boundary to impose the water levels. Note that the boundary lines must be just \emph{outside} the model. Also note that for the boundary definition when using morphology, each branch must have \emph{its own} polyline, otherwise the morphological boundary condition is invalid.}
\item{Recall that in SOBEK3 the water level at the downstream boundary holds at the last pressure point in the grid, while in \DFLOWFM it gets mirrored half a grid cell outside the control volume around each grid point. Effectively, this means that the water level has to be specified as if it is $\Delta x$ further downstream, considering the bed slope. For this case, at the downstream boundary, $\Delta x = 500 m$. For a bed level slope of \num{1d-4} this implies a 5 cm lower water level compared to SOBEK3.}
\item{In the MOR input file, the boundaries need to be imposed at the polylines specified in one of the steps above.}
\item{For the upstream (us) and downstream (ds) bed boundary condition we specify 
\begin{verbatim}IBedCond_us=1\end{verbatim} (fixed bed), and 
\begin{verbatim}IBedCond_ds=0\end{verbatim} (free condition). 
Note that, since morphological updating is switched off, this is not relevant for this case though.}
\end{itemize}

%
\begin{table}[h!]
\centering
\caption{Parameter values for the different test cases.}\label{tab:mpm_parametervalues}
\begin{tabular}{| l | l | l | l | l |}
\hline
   & \textbf{T1} & \textbf{T2} & \textbf{T3} & \textbf{T4}\\
\hline                       
{Sediment size $D_{50}$ (mm)} & 0.6 & 0.6 & 0.6 & 0.6\\
\hline
{Ch\'ezy roughness $C$ (m$^{1/2}$/s)} & 50  & 70 & 50 & 50\\
\hline
{Bed slope $i_b$ (m/m)}  & 0.0001 & 0.0001 & 0.0001 & 0.0002\\
\hline
\end{tabular}
\end{table}

% Table generated by Excel2LaTeX from sheet 'T1'
\begin{table}[htbp]
  \centering
  \caption{Model Settings}
    \begin{tabular}{l|lll}
    \textbf{Input description} & Symbol  & Value & Unit  \\
    \hline
    \textbf{flow} &       &       &  \\
    gravitational acceleration & g     & 9.81 & ms\textsuperscript{-2} \\
    branch length & s$_{tot}$  & 10000 & m \\
    height upstream crosssection & z$_{\textrm{offset}}$ & 1 & m \\
    bed level slope & i     & 0.0001 / 0.0002 & m/m \\
    cross section width & B     & 200 & m \\
    upstream discharge boundary & Q     & Q(t)  & m\textsuperscript{3}s\textsuperscript{-1} \\
    downstream water level boundary & h     & Q-h   & m \\
    Chezy roughness coefficient & C     & 50 / 70 & m\textsuperscript{1/2}s\textsuperscript{-1} \\
    initial water depth & h$_{ini}$  & 4.71 & m \\
    water density & $\rho_w$    & 1000 & kg m\textsuperscript{-3} \\
          &       &       &  \\
    \textbf{morphology parameters} &       &       &  \\
    sediment diameter & D$_{50}$   & 0.0006 & m \\
    sediment density & $\rho_s$    & 2650 & kg m\textsuperscript{-3}  \\
    relative density & $\Delta$     & 1.65 &  \\
    porosity & $\epsilon_p$    & 0.4 &  \\
    engelund-hansen calibration parameter & $\alpha$     & 1 &  \\
          &       &       &  \\
    \textbf{model parameters} &       &       &  \\
    computational grid size & $\Delta x$    & 500-2000 & m \\
    computational time step & $\Delta t$    & 60 & s \\
    output time step &       & 24 & h \\
    \end{tabular}%
  \label{tab:modeltestcase1}%
\end{table}%

\section*{Results - hydraulics, comparison SOBEK3 vs \DFLOWFM}
%
Remember that for this case, we have 3 distinct discharges, all run for 1 day (to a stationairy condition). \\
The figures below compare some hydraulic output between SOBEK3 and \DFLOWFM.

\begin{itemize}
\item{\autoref{fig:mpm_01waterlevelnode2T1}: Apart from some "initial condition effects" the upstream water level for branch T1 compares very well between SOBEK3 and \DFLOWFM.}
\item{\autoref{fig:mpm_02waterleveldifferencenode2T1}: Differences in water level just a couple of cm.}
\item{\autoref{fig:mpm_03waterlevelT1}: Also along the channels only small differences in water level.}
\item{\autoref{fig:mpm_04waterleveldifferenceT1}: Differences in water level small.} 
\item{\autoref{fig:mpm_05waterdepthT1}: Some strange differences near the boundaries of the model.} 
\end{itemize}

%
\begin{figure}%[h!]
	\includegraphics[width=\textwidth]{figures/c02_fig01_waterlevelnode2T1.png}
	\caption{Time series of water level at the second computational node at branch T1 (lowest).}
	\label{fig:mpm_01waterlevelnode2T1}
\end{figure}

\begin{figure}%[H]
	\includegraphics[width=\textwidth]{figures/c02_fig02_waterleveldifferencenode2T1.png}
	\caption{Time series of difference in water level at the second computational node at branch T1 (lowest).}
	\label{fig:mpm_02waterleveldifferencenode2T1}
\end{figure}

\begin{figure}%[H]
	\includegraphics[width=\textwidth]{figures/c02_fig03_waterlevelT1.png}
	\caption{Water level along the channel for branch T1 (top: Q1, center: Q2, bottom: Q3).}
	\label{fig:mpm_03waterlevelT1}
\end{figure}

\begin{figure}%[H]
	\includegraphics[width=\textwidth]{figures/c02_fig04_waterleveldifferenceT1.png}
	\caption{Water level difference along the channel for branch T1 (top: Q1, center: Q2, bottom: Q3).}
	\label{fig:mpm_04waterleveldifferenceT1}
\end{figure}

\begin{figure}%[H]
	\includegraphics[width=\textwidth]{figures/c02_fig05_waterdepthT1.png}
	\caption{Water depth along the channel for branch T1 (top: Q1, center: Q2, bottom: Q3).}
	\label{fig:mpm_05waterdepthT1}
\end{figure}

\section*{Results - sediment transport \DFLOWFM}
%
The differences in the sediment transport between results of \DFLOWFM and analytical values are shown in \autoref{fig:mpm_sedtr_Q1} for Q1, in \autoref{fig:mpm_sedtr_Q2} for Q2 and in \autoref{fig:mpm_sedtr_Q3} (Q3). Each line represents a branch, T1 being the most southern branch, T4 the most northern branch.
Both absolute differences \unitbrackets{kg/s} as relative differences \unitbrackets{[-]} are shown.

\begin{figure}%[H]
	\includegraphics[width=\textwidth]{figures/sedtr_Q1.PNG}
	\caption{Sediment transport from \DFLOWFM and analytical values in kg/s for Q1.}
	\label{fig:mpm_sedtr_Q1}
\end{figure}

\begin{figure}%[H]
	\includegraphics[width=\textwidth]{figures/sedtr_Q2.PNG}
	\caption{Sediment transport from \DFLOWFM and analytical values in kg/s for Q2.}
	\label{fig:mpm_sedtr_Q2}
\end{figure}

\begin{figure}%[H]
	\includegraphics[width=\textwidth]{figures/sedtr_Q3.PNG}
	\caption{Sediment transport from \DFLOWFM and analytical values in kg/s for Q3.}
	\label{fig:mpm_sedtr_Q3}
\end{figure}

The differences in sediment transport (FM1D compared to analytical results) are larger than for SOBEK3 (where the max difference was 0.000023\% for branch T2), but are acceptable small (max 1\%) and there is no structural difference. %\\
%This is still under investigation; most probably it has to do with how the Ch\'ezy coefficient is used in \DFLOWFM. \\
%Note that especially the difference in branch 2 is larger. This is because in \DFLOWFM, the global value as specified in the MDU-file is used for the sediment transport computation, which is 50 for this case. For the analytical computation, we use C=70, since that is the value for branch 2 (it is verified that for the flow it is taken into account correctly). \\
%[THIS NEEDS TO BE FIXED IN THE CODE, TO DO, note that this is also related to / linked to the division in $C_{\rm{main}}$ and $C_{\rm{floodplain}}$ for the computation of the flow velocity in the main channel (see case \ref{case:fp_Umain})]

%\begin{table}[h!]
%\centering
%\caption{Comparison of simulated sediment transport with analytical sediment transport for the different testcases.}\label{tab:mpm_result}
%\begin{tabular}{| l | l | l | l | l |}
%\hline
   %& \textbf{T1} & \textbf{T2} & \textbf{T3} & \textbf{T4}\\
%\hline                       
%{Maximum difference \unitbrackets{kg/s}} & 0.000077 &	0.000066 &	0.000038 &	0.000132 \\
%\hline
%{Maximum relative difference \unitbrackets{\%}} & 0.000024 &	0.000023 &	0.000021 &	0.000017 \\
%\hline
%\end{tabular}
%\end{table}

\section*{Conclusion}
Hydraulics almost equal between SOBEK3 and \DFLOWFM with tabulated cross sections. \\
These tests show that \DFLOWFM with tabulated cross sections is able to accurately calculate sediment formulation according to Meyer-Peter-Muller. \\

\printrefsegment
