Problem $2$

The cross$-$sectional area $A(x)$ of a vertical funnel$-$shape soil specimen of height $2ft$ in the

constant$-$head test $($see figure below$)$ varies as $A(x)=\frac{2}{x+1}$ sq $ft$ where $x($in $ft)$ is measured from

the top of the soil region. Knowing the downward flow rate $q$ must be constant at any soil cross$-$

section and the soil's average permeability coefficient $k$ is $0\mathrm{.}3f\frac{t}{sec},$ obtain

$($a$)$ the $1-$D equation of continuity with Darcy's law in terms of the total head $h(x)$ and the

unknown steady flow rate $q,$

$($b$)$ the hydraulic boundary conditions on $h(x)$ at the top $(x=0)$ and bottom $(x=2^{\text{'}})$ of the soil,

$($c$)$ the complete solution for $h(x)$ in the soil zone by applying its boundary conditions at $x=0$ and

$2^{\text{'}}$ in $($b$),$ and check that the solution satisfies the conditions of $($b$).$ Plot it over the range of $0x$

$2\text{'}$ of the soil region.

$($d$)$ the pressure elevation head $h_e(x)$ over $0x{2}^{\text{'}}$ in the soil and plot it$.$

$($e$)$ the pressure head $h_p(x)$ in the soil and plot it$.$