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80 Chapter 3 Water Sources: Groundwater
continued at the rate of pumping and a recharge well with the same flow had been superim-
posed on the discharge well at the instance the discharge was shut down. The residual draw-
down, s , can be found from Eq. 3.15 as follows:
s (114.6Q>T)[W(u) W(u )] (3.25)
where
2
u 1.87r S>4Tt
2
u 1.87r S>(4Tt )
where r is the effective radius of the well (or the distance to the observation well), t is the
time since pumping started, and t is the time since pumping stopped. For small values of r
and large values of t, the residual drawdown may be obtained from Eq. 3.19:
s (264Q>T) log (t>t ) (U.S. Customary Units) (3.26a)

Solving for T,
T (264Q>s ) log(t>t ) (U.S. Customary Units) (3.27a)
Plotting s on an arithmetic scale and t>t on a logarithmic scale, a straight line is
drawn through the observations. The coefficient of transmissivity can be determined from
the slope of the line, or for convenience, the change of residual drawdown over one log
cycle can be used as:
T 264Q> s (U.S. Customary Units) (3.28a)

where the U.S. customary units are : s (ft), Q (gpm), T (gpd/ft), t (min), t (min) and s
3
3
(ft). The comparable equations using the SI units of s (m), Q (m /d), T (m /d/m), t (min),
t (min) and s (m) are:
s¿= (0.1833Q>T) log (t>t¿) (SI Units) (3.26b)
T = (0.1833Q>s¿) log (t>t¿) (SI Units) (3.27b)
T = 0.1833Q>¢s¿) (SI Units) (3.28b)
Strictly speaking, the Theis equation and its approximations are applicable only to sit-
uations that satisfy the assumptions used in their derivation. They undoubtedly also pro-
vide reasonable approximations in a much wider variety of conditions than their restrictive
assumptions would suggest. Significant departures from the theoretical model will be re-
flected in the deviation of the test data from the type curves. Advances have recently been
made in obtaining analytical solutions for anisotropic aquifers, for aquifers of variable
thickness, and for partially penetrating wells.

3.10.4 Unconfined Aquifers

The partial differential equation governing nonsteady unconfined flow is nonlinear in h. In
many cases, it is difficult or impossible to obtain analytical solutions to the problems of
unsteady unconfined flow. A strategy commonly used is to investigate the conditions under
which a confined flow equation would provide a reasonable approximation for the head
distribution in an unconfined aquifer. These conditions are that (a) the drawdown at any
point in the aquifer must be small relative to the total saturated thickness of the aquifer, and
(b) the vertical head gradients must be negligible. This implies that the downward move-
ment of the water table should be very slow, that is, that sufficient time must elapse for the