Well Hydraulics
Part B
1
Theis Recovery Method
• Data collected during the recovery period, which
follows immediately after pumping is stopped can be
used to determine T and S as well.
• Recall, the Theis solution for transient pumping from a
well
Q∗
s(r, t) = H0 − h(r, t) =
W (u)
4πT
r2 S
u=
4T t
W (u) =
�
∞
u
e
−y
y
dy
u2
u3
u4
W (u) ≈ −0.5772 − ln(u) + u −
+
−
2 · 2! 3 · 3! 4 · 4!
2
Drawdown
3
Principle of Superposition
• states that, for all linear systems,
• the net response at a given place and time caused by
two or more stimuli is the sum of the responses
which would have been caused by each stimulus
individually.
• For wells this implies
• If two pumps operate independently in a given well,
the combined effect is the sum of the independent
effects of the two pumps
4
Representation of zero pumping
• To represent the stopping of pumping event, we
introduce a new imaginary pump that injects water to
the well at the same rate as the original pump.
• The net effect is that there is no addition or removal of
water from this time onwards.
• The solution strategy is to first find out the individual
solutions for the two pumps and then take the sum
5
Superposition
Pumping time, t
Injection time, t’
Combined effect
6
Pumping stops
Superposition of Theis Solution
Pumping
Q
W (u)
4πT
Injection
Q
�
−
W (u )
4πT
2
r S
u=
4T t
2
r S
u =
4T t�
�
Q
�
s =
{W (u) − W (u )}
4πT
�
7
Simplification of Superposition
• Use the truncated series expansion of W in the
superposition solution
2
r S
u=
4T t
W (u) ≈ −0.5772 − ln(u)
Q
�
s =
(W (u) − W (u ))
4πT
�
�
�
�
Q
T
t
T
t
�
s =
ln 2 − ln 2
4πT
r S
r S
�
�
Q
t
�
s =
ln �
4πT
t
8
Example
9
Long time solution
• After a long time of recovery, it expected that s’=0.
Therefore the solution can be re-arranged as to apply
to this time as
�
�
�
Q
Tt
Tt
0=
ln 2 − ln 2
4πT
r S
r S
�
Tt
Tt
= 2 �
2
r S
r S
t
S
=
t�
S�
10
Image Well Theory
• The original Theis solution assumes that a well is
situated within an infinite aquifer.
• Therefore, the solutions do not apply if there are
boundaries of no-flow (impermeable) or constant-head
(e.g., stream) within the zone influenced by the
pumping well.
• The superposition principle can be used to address
these principle:
• Introduce image wells at appropriate distance to
produce the desired effect of the boundaries
11
Impermeable Boundary
12
Stream
13
Image Wells Sumary
14
Image Well Solution Method
s◦ = sr + si
Q
s =
(W (ur ) − W (ui ))
4πT
�
ri2 S
ui =
ur =
4T t
4T t
� �2
ur
S
ri
=
→
u
=
u
i
r
2
rr
4T t
rr
2
rr S
rr and ri are distance of the
observation well from the
real and imaginary wells
15
Example: stream near a well
16
Example: stream near a well
17