Given: A dam which has a base width of $100m$ and upstream water depth of $70m$ is built on permeable ground layer that the thickness is $140m.$ A sheet pile extending $80m$ through the permeable soil reduces the seepage. Choose the mesh as $x=y=10m.$

Find: a$)$ Determine the potential function values. Find the velocity components under the cutoff wall taking the hydraulic conductivity of the soil as $5\mathrm{.}0{10}^{-4}\frac{m}{s}.$

b$)$ Find the seepage flow rate $(\frac{m^3}{s}\frac{?}{m})$ passing under the cutoff wall.

c$)$ Find pressures at points on the cutoff wall.

d$)$ Plot the potential and stream lines on the figure.

Governing equation for velocity potential function

$)=\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{x}},v=(\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{y}}$

Governing equation for stream function

$)=\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{y}},v=(-\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{x}}$

Boundary conditions for potential function

On $AB$ and on $AJ=1,$ on $GH$ and on $HI=0$

On BDECG $\frac{\text{d}\text{e}\text{l}}{deln}=0,$ on If $\frac{\text{d}\text{e}\text{l}}{deln}=0($ is constant on boundary$)$

Boundary conditions for stream function

On $AB$ and on $AJ\frac{\text{d}\text{e}\text{l}}{deln}=0,$

On $GH$ and on $HI\frac{\text{d}\text{e}\text{l}}{deln}=0($ is constant on boundary$)$

On BDECG $=1,$ on IJ $=0$