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

The model was developed on the premise that groundwater
flow in the East St. Louis area is two-dimensional.
The finite-difference form of the partial differential equation
(Jacob, 1950) governing the nonsteady state twodimensional
flow of ground-water is (see Stallman,
1956):
(23)
where:
h 1 = head at node 1 (see figure 60A; the aquifer is
subdivided into small squares of equal area, the
intersections of grid lines are called nodes); h 1 (i =
2, 3, 4, and 5) = heads at nodes 2 to 5; a = width
of grid interval; T = coefficient of transmissibility;
and S = coefficient of storage.
Continuing the comparison, water moves in an aquifer
just as charges move in an electrical circuit. The quantity
of water is reckoned in gallons while the charge is
in coulombs. The rate of flow of water past any point in
the aquifer is expressed in gallons per day while the flow
of electricity is in coulombs per second or amperes. The
hydraulic head loss between two points in an aquifer is
expressed in feet while the potential drop across a part
of the electrical circuit is in volts.
Thus, there are four units which are analogous;
there is necessarily a scale factor connecting each unit in
one system to the analogous unit in the other system.
Knowing the four scale factors the hydrologist is able to
relate electrical units associated with the analog model to
hydraulic units associated with an aquifer. The four
scale factors, K 1 , K 2 , K 3 , and K 4 , were defined by Bermes
(1960) as follows:
(25)
(26)
(27)
Figure 60.
Finite-difference grid (A), resistor-capacitor net (B),
and pumping rate oscilloscope trace (C)
where:
(28)
Consider a resistor-capacitor network with a square
pattern as shown in figure 60A and network junctions at
nodes as defined in figure 60B. The junctions consist of
four resistors of equal value and one capacitor connected
to a common terminal; the capacitor is also
connected to ground. The relation of electrical potentials
in the vicinity of the junction, according to Kirchhoff's
current law, can be expressed by the following
equation (see Millman and Seely, 1941; and Skibitske.
1961):
where:
(24)
Comparison of equations 23 and 24 shows that the
finite-difference equation governing the nonsteady state
two-dimensional flow of ground water in an infinite
aquifer is of the same form as the equation governing
the flow of electrical current in a resistor-capacitor network.
For every term in equation 23 there is a corresponding
term of the same order of differentiation in
equation 24.
The analogy between electrical and aquifer systems
is apparent. The hydraulic heads, h, are analogous to
electrical potentials, V. The coefficient of transmissibility,
T, is analogous to the reciprocal of the electrical resistance,
1/R. The product of the coefficient of storage, S,
and a 2 is analogous to the electrical capacitance, C.
The analogy between Ohm's law and Darcy's law is
established by the fact that the coefficient of transmissibility
is analogous to the reciprocal of the electric
resistance. Substitution of these laws in equation 27 results
in the following equation which may be used to
determine the values of the resistors of the interior portions
of the analog model (see Bermes, 1960):
where:
(30)
R = resistance, in ohms; and T = coefficient of transmissibility,
in gpd/ft.
The following equation (see Bermes, 1960), which
may be used to determine the values of the capacitors of
the interior portions of the analog model, may be derived
by taking into consideration the definitions of the coefficient
of storage and capacitance and the analogy between
(a 2 S) and C.
where:
(31)
G = capacitance, in farads; a = network spacing, in
feet; and S = coefficient of storage, fraction.
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