Ground-water development in East St. Louis area, Illinois. Urbana, IL ...

A network spacing of 1 inch equals 2000 feet was
selected to minimize the errors due to finite-difference
approximation. Equations given by Karplus (1958) suggest
that the selected network spacing is adequate.
By the process of trial and error, scale factors were
chosen so that readily available and inexpensive resistors
and capacitors and existing excitation-response apparatus
could be used.
Selected analog scale factors are given below:
A maximum pumping period, t d , of 5 years was
chosen, which is a sufficient period for water levels to
stabilize under the influence of recharge from the Mississippi
River. According to equation 28, with a K 4 =
1.826X10 5 days/sec and when t d = 5 years, the pulse
duration, t 8 , is equal to 10 -2 seconds. The pulse generator
has a maximum pulse duration of 10 -2 seconds. A scale
factor K 2 of 1 ft/volt was selected for ease in reading the
oscilloscope graph.
A generalization of equations 23 and 24 permits accounting
for variations in space of the coefficients of
transmissibility and storage by varying resistors and
capacitors. Fixed carbon resistors with tolerances of ±
10 percent and ceramic capacitors with tolerances of ±
10 percent were used in constructing the analog model.
Values of resistors were computed from equation 30
using data on the coefficient of transmissibility given in
figure 25. Values of resistors in the internal parts of
the model range in magnitude from 470,000 ohms near
the bluff where T is about 20,000 gpd/ft to 33,000 ohms
near Monsanto where T is about 330,000 gpd/ft. Resistors
are greatest in magnitude, 2,200,000 ohms, along the valley
wall where the coefficient of transmissibility is
about 5000 gpd/ft
Values of the capacitors of the interior portions of
the model were computed from equation 31 to be 2500
micro-micro farads. The long-term coefficient of storage
substituted in equation 31 was 0.15.
Excitation-Response Apparatus
The excitation-response apparatus consists of three
major parts as shown in figure 60: a waveform generator,
a pulse generator, and an oscilloscope. The waveform
generator which produces sawtooth pulses is connected
to the trigger circuits of the pulse generator and
oscilloscope, thereby controlling the repetition rate of
computation and synchronizing the oscilloscope's horizontal
sweep and the output of the pulse generator. The
pulse generator, which produces rectangular pulses of
various duration and amplitude upon command from the
waveform generator, is coupled to that junction in the
analog model representing the pumped well. The oscilloscope
is connected to junctions of the analog model where
it is desired to determine the response of the analog
model to excitation. An electron beam is swept across
the cathode ray tube of the oscilloscope providing a
time-voltage graph which is analogous to the time-drawdown
graph for an observation well. The waveform generator
sends a positive pulse to the oscilloscope to start
its horizontal sweep; at the same time, it sends a negative
sawtooth waveform to the pulse generator. At a
point along the sawtooth waveform the pulse generator
is triggered to produce a negative rectangular pulse. The
duration of this pulse is analogous to the pumping period,
t d , and the amplitude is analogous to the pumping
rate, Q. This pulse is sensed by the oscilloscope as a function
of the analog model components, boundary conditions,
and node position of the junction connected to the
oscilloscope. Thus, the oscilloscope trace is analogous to
the water-level fluctuation that would result after a step
function-type pumpage change of known duration and
amplitude. To provide data independent of the pulse
repetition rate, the interval between pulses is kept several
times the longest time constant in the analog model.
The time constant is the product of the capacitance at a
point and the resistance in its discharge path.
A means of computing the pumping rate is incorporated
in the circuit between the pulse generator and the
analog model by the small resistor, R i , in series, shown
in figure 59. Substitution of Ohm's law in equation 27 results
in the following equation which may be used to
compute the pumping rate:
where:
(32)
The voltage drop across the calibrated resistor is
measured with the oscilloscope. Switches S 1 and S 2 are
closed and opened, respectively, and the oscilloscope is
connected to the pumped well junction. The waveform in
figure 60C appears on the cathode ray tube; the vertical
distance as shown is the desired voltage drop, V R .
The switches S 1 and S 2 are returned to their original
positions. The oscilloscope is then connected to all junctions
of the analog model representing observation wells.
The screen of the oscilloscope is accurately calibrated so
that voltage and time may be used on the vertical and
horizontal axis, respectively. The time is in seconds;
the value of each horizontal division on the screen is
determined by noting the duration of the rectangular
pulse and the number of divisions covered by the timevoltage
trace for a junction adjacent to the pumped well.
The time-voltage graphs obtained from the oscilloscope
can be converted into time-drawdown graphs with equa-
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