$(25\%)$ Problem $1.$ The laboratory apparatus shown in Figure P$1$ maintains a constant head

in both the upper and lower reservoirs. The soil sample is a silty sand $($SM$)$ with $k=5x$

${10}^{-4}c\frac{m}{s},$ a moisture content of $w=17\%,$ and a specific density of $G_s=2\mathrm{.}65$ placed within

a glass tube. There is a screen on the right$-$hand side of the soil.

a$)$ What are Darcy's and seepage velocities within the soil in $c\frac{m}{s}?$

b$)$ What is the discharge through the soil in $c\frac{m^3}{s}?$

c$)$ Determine the time required for the entire plug of colored water to break through

the soil $($i$.$e$.,$ from when the leading edge of the plug of colored water enters the soil

to when the entire plug exits the soil$).$

d$)$ What is the horizontal reaction force of the screen placed on the right$-$hand side of

the soil? Use the internal approach to find the force.

Clarification: The colored water$-$plug travels only through advection and not diffusion; it

has the same viscosity as water; and there is no adsorption or retardation of the colored

water $($i$.$e$.,$ it travels as fast as water$).$