(Solved): A railroad car filled with fish poison has derailed in the Mad River valley near a bridge and much ...

A railroad car filled with fish poison has derailed in the Mad River valley near a bridge and much of the chemical has spilled into the water. An earth dam, as depicted below, is constructed to hold back contaminated water from downstream habitats to try and save some fish. It is made of clay material with a low saturated hydraulic conductivity of $\mathrm{K}=10^{?6}\mathrm{m}{\sec }^{?1}$ and high porosity $\mathrm{n}=0.5$. The water levels ${\mathrm{H}}_{\mathrm{o}}$ and ${\mathrm{H}}_1$ are kept at $2\mathrm{m}$ and $0.5\mathrm{m}$, respectively, across the length $\mathrm{L}=$ $5\mathrm{m}$ of the dam. The water in the dam can be considered to form a steady unconfined aquifer. Assume $\mathrm{R}=0$. a. Find an expression for the head profile in the dam, $h(x)$. b. Estimate the discharge per unit length of the dam. c. The travel time of contaminated water from the holding basin through the dam (distance L) can be expressed as $t_L=\frac{\text{ distance }}{\text{ velocity }}=\frac{L}{v}$ if the pore velocity is constant over the distance L. If it varies, then $t_L={?}_0^L\frac{1}{v}dx$ Estimate the travel time of contaminated water from the holding basin through the distance $\mathrm{L}$ of the earth dam.