$6.$ Consider a sloping soil overlying an impervious layer. A shallow lake at the upper end the slope recharges the soil causing saturated flow down$-$slope to a seepage face $($a cliff face$).$ A piezometer tube near the upper end of the slope measures a soil water pressure of $30$ cm $.$ The saturated hydraulic conductivity of the soil is believed to be $\backslash (1\mathrm{.}2\backslash$mathrm$\{$~cm$\}/\backslash$mathrm$\{$hr$\}\backslash ).$

a$).$ Ignoring soil water losses to evaporation and plant uptake, calculate the steady$-$state soil water flux at the seepage face.

b$).$ Suppose that the lake level fluctuates seasonally resulting in a changing soil water pressure at the upper end of the slope. The time dependence in the water pressure, $\backslash (\backslash$mathrm$\{$P$\}(\backslash$mathrm$\{$cm$\})\backslash ),$ from May $1$ thru Sept. $1$ is described by

$\backslash [$

$\backslash$mathrm$\{$P$\}(\backslash$mathrm$\{$t$\})=30+10\backslash$sin $(2\backslash$pi $\backslash$mathrm$\{$t$\}/120)$

$\backslash ]$

where $\backslash ($ t $\backslash )$ is in days and the argument of the sin term is in radians. Calculate and plot $\backslash ($ q$($t$)\backslash )$ at the seepage face for $\backslash ($ t$=0\backslash )$ to $\backslash ($ t$=120\backslash )$ days. For simplicity, you may assume that steady state flow is quickly achieved at each new pressure through the season, and you may also ignore temperature change.

c$).$ The average velocity of the soil water moving downslope in the saturated zone is $\backslash (\backslash$mathrm$\{$V$\}\_\{\backslash$mathrm$\{$w$\}\}=\backslash$mathrm$\{$q$\}/\backslash$theta$\_\{\backslash$mathrm$\{$s$\}\}\backslash ).$ If the soil has an average bulk density of $\backslash (1\mathrm{.}4\backslash$mathrm$\{$~g$\}\backslash$mathrm$\{$~cm$\}^\{-3\}\backslash ),$ estimate the average time of transit for water moving from the lake to the seepage face when $\backslash (\backslash$mathrm$\{$P$\}=30\backslash$mathrm$\{$~cm$\}\backslash ).$

d$).$ The calculations thus far required an estimate of Ks$.$ Suppose that Ks is not known. To estimate $\backslash (\backslash$mathrm$\{$K$\}\_\{\backslash$mathrm$\{$s$\}\}\backslash ),$ you apply a water tracer $($e$.$g$.$ a non$-$adsorbing solute$)$ to the saturated soil at the upper end of the slope and observe its average arrival time at the seepage face is $15$ days after application. Calculate the soil Ks$.$