\chapter{Use of WBI piping kernel} \label{sec:UseWBIPipingKernel}

Functional designs
The functional design of the WBI piping kernel is described in \citep{PipingKernel_FunctionalDesign}.

The WBI use of the piping kernel consists of three sub failure mechanisms: uplift, heave and backward erosion. 
The complete calculation is done by:
\begin{enumerate}
	\item The calculation of the uplift safety by determining the vertical balance of weight of the subsoil and the waterpressure at the top of the aquifer.
\item The calculation of heave by checking the maximal gradient over the vertical waterflow at the uplift location. Heave is the vertical sand transport through the horizontal pipes towards the location of uplift breaching (the exit location.)The thickness layer is the distance where over heave occurs.
\item The calculation of internal erosion with Sellmeijer revised.
%\item Checking the presence of sufficient horizontal seepage length (kwelweglengte) 
%\item Determiniation of the piping safety factor by dividing the required seepage length by the present seepage length.
\end{enumerate}

The use by DAM of these functions is described in following paragraphs.

\section{Uplift (uplift safety)}\label{sec:Uplift}
For the uplift calculation DAM uses the DAM uplift calculation described in \autoref{sec:UpliftCalculation}

%This function of the kernel is decribed in paragraph 3.3 in \citep{PipingKernel_FunctionalDesign}.
%
%Input of the kernel consists of:
%
%\begin{table}[H]
	%\centering
		%\begin{tabular}{|p{20mm}|p{20mm}|p{50mm}|p{50mm}|} \hline
%\textbf{Symbol}& \textbf{Unit}  & \textbf{Description} & \textbf{Value in DAM}  \\ \hline
%h$_{exit}$		 & m	 & phreatic level at the exit point (above reference level NAP)     & calculated, see \autoref{sec:PiezHeadUpliftLocation}    \\ \hline
%$\Phi _{polder}$ & m & piezometric head in the hinterland (above reference level NAP) & \textit{\textcolor[rgb]{0.65,0.16,0}{HeadPl2}}\ \\ \hline
%$\Phi _{exit}$         & m     & piezometric head at the exit point (above reference level NAP)  & calculated, see \autoref{sec:PiezHeadUpliftLocation} \\  \hline
%r$_{exit}$               &       & damping factor at the exit point  &  calculated, see \autoref{sec:PiezHeadUpliftLocation}  \\ \hline
%m$_{u}$        & -   & model factor uplift &   ?    \\ \hline
%D$_{cover,i}$	    & m    & thickness of the cover sublayer i at the exit point &  calculated, see \autoref{sec:DeterminationSoilParameters}   \\ \hline
%$\sigma_{eff}$ &	kN/m$^{2}$ & effective vertical stress at the bottom of the cover layer & calculated, see \autoref{sec:DeterminationSoilParameters}  \\ \hline
%$\gamma_{eff,cover,i}$   & kN/m$^{3}$ & effective volumetric weight of cover sublayer i  & calculated, see \autoref{sec:DeterminationSoilParameters} \\ \hline
%$\gamma_{water}$  & kN/m$^{3}$  & volumetric weight of water &  9.81  \\ \hline
%$\gamma$          & m     & leakage length on the landside of the dike (hinterland)    & New input for DAM \\ \hline
		%\end{tabular}
	%\caption{Input paramaters}
	%\label{tab:InputParametersUplift}
%\end{table}
%
%Output of the kernel for the uplift safety calculation is:
%\begin{itemize}
	\item Z$_u$ (limit state function value)
	\item FoS$_{u}$ (factor of safety)
	\item $\Delta \Phi _{c,u}$ (critical head difference for uplift)
	\item h$_{c,u}$(critical water level for uplift)
	\item D$_{cover,i}$ (effective thickness of the cover layer at exit point)
	\item $\gamma _{eff}$(effective stress at the exit point)
	\item h$_{exit}$(piezometric head at the exit point)
\end{itemize}


\section{Heave} \label{sec:Heave}
This function of the kernel is decribed in paragraph 3.4 in \citep{PipingKernel_FunctionalDesign}.

In DAM is assumed that heave always occurs, until the connection to the WBI piping kernel is extended to the complete version (also adeptions in DAM UI)

%Input of the kernel consists of:
%
%\begin{table}[H]
	%\centering
		%\begin{tabular}{|p{20mm}|p{20mm}|p{50mm}|p{50mm}|} \hline
%\textbf{Symbol}            & \textbf{Unit}  & \textbf{Description}    &\textbf{Value in DAM   }            \\ \hline
%i                 & -     & gradient at exit point & calculated based on the damping factor, see \autoref{sec:PipingWaterpressures} \\ \hline
%i$_{c,h}$                & -     & critical exit gradient & calculated based on the damping factor\\ \hline
%D$_{cover}$	    & m     & total thickness of the cover sublayer &  calulated, see \autoref{sec:DeterminationSoilParameters}   \\ \hline	
%h$_{exit}$ & m NAP & piezometric head at the exit point & output kernel 	\\ \hline	
%$\Phi _{polder}$ & m & piezometric head in the hinterland (above reference level NAP) & c\textit{\textcolor[rgb]{0.65,0.16,0}{HeadPl2}}\\ \hline	
		%\end{tabular}
	%\caption{Input paramaters Heave}
	%\label{tab:InputParametersHeave}
%\end{table}
%
%Output of the kernel for the heave calculation is:
%\begin{itemize}
	%\item Z$_h$ (limit state function value)
	%\item FoS$_{h}$ (factor of safety)
	%\item h$_{c,h}$(critical water level for heave)
	%\item h$_{exit}$(piezometric head at the exit point)
	%\item i (gradient at exit point)
%\end{itemize}

\section{Internal erosion (backward erosion)}\label{sec:InternalErosion}

The WBI piping kernel facilitates the models Bligh, Sellmeijer both in original as revised (WTI2011)form. For now the use by DAM is restricted to Sellmeijer revised (WTI 2011)
This function of the kernel is decribed in paragraph 3.5 in \citep{PipingKernel_FunctionalDesign}.

Input of the kernel consists of:

\begin{table}[H]
	\centering
		\begin{tabular}{|p{20mm}|p{20mm}|p{50mm}|p{50mm}|} \hline
\textbf{Symbol}& \textbf{Unit}  & \textbf{Description} & \textbf{Value in DAM}  \\ \hline
h              & m   & river water level (above reference level NAP) &\textit{\textcolor[rgb]{0.65,0.16,0}{BoezemLevelTp}} or WaterHeight (when using scenarios) \\ \hline
h$_{exit}$		 & m	 & phreatic level at the exit point (above reference level NAP)     & calculated, see \autoref{sec:PhreaPlane} \\ \hline
m$_{p}$        & -   & model factor piping &   1.0    \\ \hline
$\gamma_{water}$  & kN/m$^{3}$  & volumetric weight of water &  9.81  \\ \hline
r$_{c}$				 & -   & reduction factor providing the fraction of the blanket thickness by which the total head difference is reduced due to hydraulic resistance in the vertical exit channels & 0.3 \\ \hline
D$_{cover}$    & m     & total thickness of the cover layer at the exit point   & calculated, see \autoref{sec:DeterminationSoilParameters} \\ \hline
$\gamma_{sub,particals}$ & kN/m$^{3}$  & submerged volumetric weight of sand particles   &       16.5                               \\ \hline
$\theta_{Sellmeijer,rev.}$  & deg   & bedding angle for Sellmeijer original &      37                                               \\ \hline
$\eta$                   & -     & White’s drag coefficient        &   0.25                                         					      \\ \hline
d$_{70}$                 & m     & 70\%-fractile of the aquifer’s grain size distribution   &  from soilmaterials.mdb               \\ \hline
d$_{70m}$                & m     & d70-reference value in Sellmeijer, revised    &    2.08E-4                                       \\ \hline
$\kappa$                 & m$^{2}$     & intrinsic permeability      & calculated with  k, $\nu_{water}$ and g                      \\ \hline
k                        & m/s         & hydraulic conductivity (Darcy)          & from soilmaterials.mdb                           \\ \hline
$\nu_{water}$          & m$^{2}$ /s  & kinematic viscosity of water at 10 degrees Celsius   &  1.33 E-6                             \\ \hline
g                      & m/s$^{2}$   & gravitational constant       & 9.81                                                          \\ \hline
D				  			       & m		 & thickness of the aquifer	& calculated, see \autoref{sec:DeterminationSoilParameters}  							\\ \hline
L                      & m     & seepage length                 &  calculated, see \autoref{sec:DeterminationSoilParameters}        \\ \hline
		\end{tabular}
	\caption{Input paramaters Internal erosion}
	\label{tab:InputParametersInternalErosion}
\end{table}

Output of the kernel for the internal erosion calculation is:

\begin{itemize}
	%\item Z$_p$ (limit state function value)
	\item FoS$_{p}$ (factor of safety)
	%\item h$_{c,p}$(critical water level for uplift)
	%\item $\Delta$ H$_{c}$ (critical head difference)
	%\item h - $_{exit}$- r$_{c}$D$_{cover}$(reduced head drop this the head drop which is reduced by the head drop over the exit channel)
\end{itemize}



%\section{Piezometric head and uplift location}\label{sec:PiezHeadUpliftLocation}
%The kernel needs the piezometric head in the aquifer. Herefore the demping factor r$_{toe}$ is required input. This is the damping factor defined by formula 3.1 in functional design kernel. In the WBI software this is an user defined input paramater. In DAM r$_{toe}$ can be calculated from the DAM input \textit{\textcolor[rgb]{0.65,0.16,0}{DampingFactorPl3}} (or \textit{\textcolor[rgb]{0.65,0.16,0}{DampingFactorPl4}} if PL4 is present): 
%
%\begin{align} \label{eq_dempingsfactor}
%% if PL2 and Phi polder are not the same r_{toe}= \frac{(h- (\textit{\textcolor[rgb]{0.65,0.16,0}{DampingFactorPl3}} * (h - \textit{\textcolor[rgb]{0.65,0.16,0}{HeadPl2}})))-\Phi _{polder}}{ h - \Phi _{polder}}
%r_{toe} = 1 - \textit{\textcolor[rgb]{0.65,0.16,0}{DampingFactorPl3}}
%\end{align}
%
%In the WBI piping kernel is an assumption made for the distribution ot the piezomeric head in the aquifer at different x locations. This requires a leakage length $\lambda$. This is a user defined parameter (new to DAM).
%Parameter used in DAM is \textit{\textcolor[rgb]{0.65,0.16,0}{SlopeDampingPiezometricHeightPolderSide}}. remark Irene (2-5-2018): Perhaps it is an option to calculate the leakage length from this parameter to avoid adding new parameters to DAM. This belongs to the extended version of the WBI-piping connection, as adaption in the UI is needed.
%
%
%\section{Determination of soil parameters}\label{sec:DeterminationSoilParameters}
%DAM already provides soil related parameters such as thickness of cover layer, position aquifer,etc. for the DAM piping calculations.
%These schematisations are also used for the WBI-piping kernel.
%remark Irene (2-5-2018): still needs to be described in FO (MWDAM-1252-priority:minor).
