[round off to 2 decimal places] (Take $\pi = 3.14$)
Steady State Pumping Test Drawdown Analysis
This problem requires calculating the drawdown in a main pumping well under steady-state flow conditions, using data from observation wells at known distances.
Drawdown Behavior in Steady State
For steady-state radial flow in a confined aquifer, the drawdown ($s$) is related to the distance ($r$) from the pumping well. This relationship is linear when plotted against the natural logarithm of the distance ($\ln(r)$).
The formula can be expressed as:
$s(r) = A \ln(r) + B$
where $A$ and $B$ are constants derived from the aquifer properties and pumping rate. We can determine the drawdown in the main well ($s_w$) at its radius ($r_w$) by using the drawdowns measured at two observation wells ($r_1, s_1$) and ($r_2, s_2$).
Using Observation Well Data
Given information:
- Observation Well 1: $r_1 = 30 \ m$, $s_1 = 1.11 \ m$
- Observation Well 2: $r_2 = 90 \ m$, $s_2 = 0.53 \ m$
- Main Well Diameter: $D = 30 \ cm = 0.3 \ m$. Thus, the main well radius is $r_w = D/2 = 0.15 \ m$.
Calculating Drawdown in Main Well
We can use a linear interpolation method based on the $\ln(r)$ scale to find $s_w$. The formula to find $s_w$ at $r_w$ using two points $(r_1, s_1)$ and $(r_2, s_2)$ is:
$s_w = s_1 + (s_2 - s_1) \frac{\ln(r_w / r_1)}{\ln(r_2 / r_1)}$
Substituting the given values:
$s_w = 1.11 + (0.53 - 1.11) \frac{\ln(0.15 \ m / 30 \ m)}{\ln(90 \ m / 30 \ m)}$
$s_w = 1.11 + (-0.58) \frac{\ln(1/200)}{\ln(3)}$
Using the property $\ln(1/x) = -\ln(x)$:
$s_w = 1.11 - 0.58 \frac{-\ln(200)}{\ln(3)}$
$s_w = 1.11 + 0.58 \frac{\ln(200)}{\ln(3)}$
Numerical Evaluation
Now, we evaluate the expression using approximate values for the natural logarithms:
- $\ln(3) \approx 1.0986$
- $\ln(200) \approx 5.2983$
$s_w \approx 1.11 + 0.58 \times \frac{5.2983}{1.0986}$
$s_w \approx 1.11 + 0.58 \times 4.8226$
$s_w \approx 1.11 + 2.7971$
$s_w \approx 3.9071 \ m$
Final Result
Rounding the calculated drawdown to 2 decimal places as required:
Drawdown in the main well = 3.91 m
The provided correct value lies between 4 and 4.25.
Important Questions from Steady Flow Through Wells
- A well fully penetrates a $30\text{ m}$ thick confined aquifer. After a long period of pumping at a constant rate of $0.06\text{ m}^3\text{.s}^{-1}$, drawdowns of $3.6\text{ m}$ and $2.0\text{ m}$ are found at distances $60\text{ m}$ and $120\text{ m}$ away from the well, respectively. The hydraulic conductivity of the aquifer is $n \times 10^{-4}\text{ m.s}^{-1}$. The value of $n$ is ________ (Rounded off to two decimal places)
- A $40$ cm diameter tubewell is constructed in a $10$ m thick confined aquifer having hydraulic conductivity of $25$ m day$^{-1}$. The peizometric surface is observed to be $40$ m high from the impervious stratum at the radius of influence of $500$ m. The drawdown in the tubewell is $30$ m. If the thickness of aquifer is doubled and diameter of tubewell is reduced to half, keeping all other parameters and conditions same, the change in discharge from the well is [Take $ \pi = 3.14$]
- In an area, one tubewell of 15 cm diameter was operating such that groundwater table is 40 m above the bottom of screen. The groundwater formation has permeability (K) as 20 m.d$^{-1}$ and well has radius of influence of 980 m with a maximum drawdown of 12 m. After some time, a second tubewell of same size was installed at a distance of 500 m from the first tubewell. If both the tubewells are running simultaneously and have equal discharge, the loss of discharge in the first tubewell due to installation of the second tubewell, in %, is _________. (Rounded off to 2 decimal places)
- A $0.30$ m diameter well penetrates an unconfined aquifer with a saturated depth of $40$ m. After $8$ hours of pumping at a steady rate of $0.03$ $m^3$ $s^{-1}$, the drawdown in two observation wells located at $20$ m and $50$ m away from the pumping well are found to be $3$ m and $2$ m, respectively. The drawdown in the pumping well in m is ________ (rounded off to $1$ decimal place, Consider $\pi = 3.14$).
Water from a confined aquifer having transmissivity of 1000 $m^2 day^{-1}$ is pumped through a fully penetrating well of 300 mm diameter at a rate of 1200 $m^3 day^{-1}$. If the radius of influence is 400 m, the drawdown in the well under steady-state flow condition in meter is __________ [round off to two decimal places]
(Take $\pi = 3.14$)
