In a falling head test, the initial head at $t=0$ is $72$ cm $.$ At $t=30$min, the head is $66$ cm $.$ The

diameters of the standpipe and the specimen are $1\mathrm{.}5$ and $25$ cm $,$ respectively. The length

of the specimen is $25$ cm $.$ Calculate the hydraulic conductivity $($in $\frac{m}{s})$ and intrinsic

permeability $($in $m^2)$ of the specimen. Assume, density of water as $998\mathrm{.}29kg{m}^{-3}$ and the

dynamic viscosity as $1\mathrm{.}003{10}^{-3}kg{m}^{-1}{s}^{-1})$

$(10$ points, $30$ min $)$

The $2-$h unit hydrograph for a watershed is given below. The losses are $10m\frac{m}{h}$ and $5$

$m\frac{m}{h}$ for first and second storm, respectively.

If the base$-$flow is assumed to be $10\frac{m^3}{s}$ in the beginning and increased by $2\frac{m^3}{s}$ every

hour, compute the following:

$($a$)$ The flood hydrograph $($in $\frac{m^3}{s})$

$($b$)$ Area of the watershed $($in $k{m}^2)$

$(10$ points, $30$ min $)$

A bridge is planned to be constructed at the outlet of a basin for a peak flood of $20-$year

return period. The basin upstream of this location has an area of $62k{m}^2.$ The maximum

length of the main stream is $10$ km $,$ and the basin has the average slope of $1\mathrm{.}3\%.$ The

depth of the rainfall in the basin for a return period of $20-$year is given by $h=28\mathrm{.}6t^{0\mathrm{.}416},$ where

$h$ is in $\text{'}m{m}^{\text{'}}\text{'}$ and $t$ is the rainfall duration in $\text{'}h\text{'}.$ Estimate the design flow $($in $\frac{m^3}{s}).$ The runoff

coefficient can be taken as $0\mathrm{.}2.$

$(10$ points, $30$ min $)$

A flood control reservoir was designed for the N $-$year flood. Assuming that the successive

annual floods are independent, calculate the following:

$($a$)$ What is the probability that a flood equal to or greater than the $50-$year flood will occur

in $50$ years?

$($b$)$ What is the probability that one or more floods equal to or greater than the $50-$year will

occur in $50$ years?

$($c$)$ If a company constructs $20$ independent systems $-$ i$.$e$.,$ in $20$ different areas $-$ designed

for the flood of $500$ years, what is the probability that $3$ of the systems fail at least once

in the first $50$ years after the construction.

$(10$ points, $30$ min $)$

An irrigation project plans to divert a water from a nearby river to irrigate the agricullunal

land in the downstream. In order to assess the availability of water during the dry seasion,

low flow analysis has to be carried out. The table below provides the annual minimum

flows recorded just upstream of the proposed intake location. An environmental flow of at

least $10\frac{m^3}{s}$ is required to be released downstream.

If the project will be designed for $20-$year return period flow, estimate the design flow by

fitting data in Log$-$normal distribution.