Groundwater wells are used for a large number of purposes in the environmental and geosciences fields:

-extract water for municipal, domestic, industrial and irrigation use
-control salt water intrusion
-remove contaminated water during site cleanups
-control the flow of uncontaminated water in the vicinity of contaminated areas
-dewater during construction activities.

In an aquifer, the height to which water rises in a well above some common datum is referred to as the hydraulic head, represented by h When wells are pumped, the hydraulic head in the pumped aquifer drops in response to the pumping. The difference in the hydraulic head prior to pumping to that after pumping has been in progress for a period of time is referred to as the drawdown, represented by s. Assuming a horizontal head distribution prior to pumping, then

s = h₀ - h                       Eq A

where s is the drawdown as a function of distance, r, from the pumping source

h is the hydraulic head as a function of distance, r, from the pumping source

h_o is the constant hydraulic head in the aquifer prior to pumping

In general, drawdown is dependent upon radial distance, r, from the pumping source, and the time since pumping began, t. That is drawdown, s is a function of r and t.

However, if a well has been pumped for a significant period of time, the rate of increase in drawdown will approach zero at any distance r from the pumping well so that eventually pumping will only minimally affect drawdown.

At this point "quasi-steady state" conditions have been achieved and the hydraulic head, and drawdown, are considered to be functions of only the radial distance, r, from the pumping source.

A. Use Eq A to find the relationship between dh/dr and ds/dr.

B. Under quasi-steady state conditions, the following first order differential equation relates the hydraulic head, h, in the pumped confined aquifer as a function of radial distance, r, from the pumping source:

Q/2 π T r = dh/dr                 Eq B

Where h is the hydraulic head as a function of distance, r, from the pumping source

r is the radial distaqnce from the pumoing source

q is the constant pumping rate of the well (L³/time)

T is an aquifer property called the transmissivity, here it is considered a constant (L²/time)

Use your result from Part A to rewirte Eq B so it involves ds/dr instead of dh/dr. Then the general solution to this modified Eq B for a as a function of r.

C. Find the particular solution subject to the initial condition that the drawdown at some given distance r = R from the pumping source is a value of s_R, that is s(r) = s_R.

D. A well in a confined aquifer with a transmissivity of 690 m²/day is pumped at 2700 m³/day until quasi-steady state conditions have been reached. At that point, drawdown in an observation well 50 meters from the pumping source is 0.80 meters. Find the drawdown at a distance 100 meters from the pumping source.

E. Graph the drawdown at 5 meter increments for 5 ≤ r ≤ 100.