$3.[30$ points$]$ To build an underwater foundation, a temporary sheetpile wall system has been constructed as shown in the Figure below, and the soil has been excavated to a depth of $\backslash (\backslash$mathrm$\{$D$\}=6\backslash$mathrm$\{$~m$\}\backslash ).$ The water level H on the back side of sheet pile is $8$ m $.$ The corresponding flow net for this problem is also shown in figure below.

a$)$ At what rate is water being pumped out of the excavation to maintain the water level shown?

b$)$ What is the vertical effective stress at point A $?$

c$)$ What is the factor of safety against the heaving in the critical regions around the sheet piles? $($Hint: calculate the hydraulic gradient of the cell highlighted in yellow; $\backslash ($ F S$=\backslash )\backslash (\backslash$frac$\{$i$\_\{$c r$\}\}\{$i$\}$ ; i$\_\{$c r$\}=\backslash$frac$\{\backslash$gamma$^\{\backslash$prime$\}\}\{\backslash$gamma$\_\{$w$\}\}\backslash ))$

d$)$ How high H would water have to be on the back side of the sheet pile wall to create an unstable situation $($i$.$e$.,$ heaving is triggered$)$ in the critical regions? $($Assume that the water level in the excavation remains as shown in the Figure.$)$