Figure below shows a $1-$D transient groundwater problem. Identify the governing equation and initial and boundary conditions for this problem.

$L=$ TUU $m$

Analytical solution to this problem can be written as:

$h(x,t)=h_1+\frac{(h_2-h_1)x}{L}+\frac{2}{}[{\displaystyle \underset{n=1}{\overset{}{}}}\frac{(h_2-h_1)cos(n)}{n}sin(\frac{nx}{L})exp(-T{n}^2{}^2\frac{t}{S}{L}^2)]$

Develop solution procedure $($in MATLAB, EXCEL, VB$,$ FORTRAN, OR C$)$ and generate the solution at every $10$ meter $)$ for various time $=10$mins,$20$mins,$40$mins,$80$mins,$160$mins, $250$mins,$400$mins, and $500$mins. Plot these results. Use a maximum of $10$ terms in the infinite series. If you are using pure EXCEL spreadsheets then just use $3$ terms to keep it simple.

Derive the steady state solution for this system. Compare you steady$-$state solution based results to the results based on the numerical solution. Also, estimate how long it took to reach steady state?

Perform following sensitivity analysis:

What happens to the solution if S value is decreased to $0\mathrm{.}001?$

What if $T$ is increased to $0\mathrm{.}04?$

Does the time taken to reach steady state changed when you changed the model parameters?

Discuss your results

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