Chapter 12:Variable-discharge tests and tests
in well fields
•Aquifers may be pumped at variable discharge
rates either deliberately or due to characteristics of
the pump.
•Aquifers can be pumped step-wise (always pumped
but pumping rates vary) or may be pumped
intermittently (not always pumped and pumping
rates can vary).
Variable-discharge tests
Confined Aquifers, Birsoy-Summer’s Method
Birsoy and Summers present an analytical
solution for the drawdown response in a
confined aquifer that is pumped step-wise or
intermittently .
 They apply the principle of superposition (ch.6)
to Jacob’s approximation of the Theis equation
(3.7) (shown below)

Variable-discharge tests
Confined Aquifers, Birsoy-Summer’s Method

The drawdown in the aquifer at time t during the
nth pumping period of intermittent pumping
is shown by the following expression:
where
where
where
Intermittent Pumping
Variable-discharge tests
Confined Aquifers, Birsoy-Summer’s Method

For step-wise (uninterrupted pumping):
t’(i-1) = ti, and the ‘adjusted time’
where
becomes
Variable-discharge tests
Confined Aquifers, Birsoy-Summer’s Method

If the intermittent pumping rate is
constant (Q=Q1 =Q2=…….Qn) then the
adjusted time becomes:
Dividing both sides of equation 12.1 by Qn
gives an expression for drawdown:
Step-wise Pumping
Variable-discharge tests
Confined Aquifers, Birsoy-Summer’s Method
Assumptions (from Ch.3):
1.)The aquifer is confined
2.)The aquifer has a seemingly infinite areal extent
3.)The aquifer is homogeneous, isotropic, and of uniform thickness
over the area influenced by the test
4.)Prior to pumping, the piezometric surface is horizontal (or
nearly so) over the area influenced by the test
5.)The aquifer is pumped step-wise or intermittently at a
variable discharge rate or is intermittently pumped at a
constant discharge rate
6.)The well penetrates the entire thickness of the aquifer and thus
receives water by horizontal flow
The following conditions are added:
the flow to the well is in an
unsteady state
R is small and t is sufficiently large
Variable-discharge tests
Confined Aquifers, Aron-Scott’s Method
•The sharpest decrease in discharge occurs
soon after the start of pumping. Aron and
Scott take this into account. They show
that when:
where sn=drawdown at a certain moment tn,; se=excess drawdown caused by the
earlier higher discharge
Variable-discharge tests
Confined Aquifers, Birsoy-Summer’s Method
•Determine the slope of the straight line
Variable-discharge tests
Confined Aquifers, Aron-Scott’s Method
If the fully developed drawdown is
considered to extend to the
distance ri at which
then the se (excess
drawdown) can be
approximated by:
Variable-discharge tests
Confined Aquifers, Aron-Scott’s Method
Assumptions:
All same as in chapter 3, except:
5.)The discharge rate decreases with
time, the sharpest decrease
occurring soon after pumping
the following condition is added:
Free-Flowing Wells
Based on the conditions that the
drawdown in the well is constant and
discharge decreases with time.
 To satisfy these conditions, the well is
shut until pressure becomes static, then at
t=0 the well is opened and the water level
in the well drops instantaneously to a
constant drawdown level which is equal
to the outflow. The well discharges at a
decreasing rate.

Free-flowing wells
Confined aquifer, unsteady-state flow,
Hantush’s Method
where
Free-flowing wells
Confined aquifer, unsteady-state flow,
Hantush’s Method
Assumptions
All same as in chapter 3, except:
5.)At the start of the test (t=0), the
water level in the free-flowing well
drops instantaneously. At t>0, the
drawdown in the well is constant,
and its discharge is variable.
The following condition is added:
the flow to the well is in an unsteady state
Free-flowing wells
Leaky aquifer, steady-state flow,
Hantush-DeGlee’s method
where
Assumptions
All assumptions that underlie the standard
methods for leaky aquifers, except:
5.)At the beginning of the test (t=0),
the water level in the well drops
instantaneously. At t>0, the
drawdown in the well is constant,
and it’s discharge is variable.
Well-fields