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72 Chapter 3 Water Sources: Groundwater
amount equal to the rate of withdrawal from the well. A steady state is then reached and
the water level ceases to decline.

3.10 NONSTEADY RADIAL FLOW
Solutions have been developed for nonsteady radial flow toward a discharging well.
Pumping test analyses for the determination of aquifer constants are based on solutions of
unsteady radial flow equations.

3.10.1 Confined Aquifers

In an effectively infinite artesian aquifer, the discharge of a well can only be supplied through
a reduction of storage within the aquifer. The propagation of the area of influence and the rate
of decline of head depend on the hydraulic diffusivity of the aquifer. The differential equation
governing nonsteady radial flow to a well in a confined aquifer is given by
2
0 h 1 0h S 0h
+ = (3.11)
2
0r r 0r T 0t
Using an analogy to the flow of heat to a sink, Theis (1963) derived an expression for
the drawdown in a confined aquifer due to the discharge of a well at a constant rate. His
equation is really a solution of the Eq. 3.11 based on the following assumptions: (a) The
aquifer is homogeneous, isotropic, and of infinite areal extent; (b) transmissivity is con-
stant with respect to time and space; (c) water is derived entirely from storage, being re-
leased instantaneously with the decline in head; (d) storage coefficient remains constant
with time; and (e) the well penetrates, and receives water from, the entire thickness of the
aquifer. The Theis equation may be written as follows:
q -u
Q e Q
s = h - h = r S du = W(u) (SI Units) (3.12)
2
0
4pT L 4Tt u 4pT
where h is the head at a distance r from the well at a time t after the start of pumping; h is
0
the initial head in the aquifer prior to pumping; Q is the constant discharge of the well; S is
the storage coefficient of the aquifer; and T is the transmissivity of the aquifer. The integral
in the above expression is known as the exponential integral and is a function of its lower
limit. In groundwater literature, it is written symbolically as W(u), which is read “well
function of u,” where

2
u = (r S)>(4Tt) (SI Units) (3.13)
The drawdown s (m) at a distance r (m) at time t (days) after the start of pumping for
3
3
a constant discharge Q (m /d) under the transmissivity T (m /d/m) is given by Eq. 3.12 and
Eq. 3.13.
Its value can be approximated by a convergent infinite series:
2 3
W(u) =-0.5772 - ln u + u - u >2 * 2! + u >3 * 3! Á (3.14)
Values of W(u) for a given value of u are tabulated in numerous publications. A partial
listing is given in Table 3.2.
The drawdown s (ft), at a distance r (ft), at time t (days) after the start of pumping for
a constant discharge Q (gpm), is given by
s = h - h = 1,440 QW(u)>(4pT) = 114.6 QW(u)>T (US Customary Units) (3.15)
0