Transcript Image Well Theory

Groundwater Pollution
Remediation
NOTE 4
Superposition
Principle of Superposition
• If Φ1 = Φ1 (x, y, z, t) and Φ2 = Φ2 (x, y,
z, t) are two general solutions of a
homogenous linear partial differential
equation L(Φ) = 0, then any linear
combination Φ = C1 Φ1 + C1 Φ2 where
C1, C2 are constants is also a solution
of L(Φ) = 0.
• Applications: multiple well systems,
non-steady pumpage, boundary
problems
Image Well Theory
(1) Barrier Boundary
d
Q
Pumping
Well
dA/dX = 0 (no flux B.C.) at X =0
x
No barrier aquifer
Q
X
r
Image Well Theory
(1) Barrier Boundary: How to compute
drawdown at the observation well?
Q
d
Image
Well
Q
d
r2
Pumping
Well
r1
Observation Well
dA/dX = 0 (no flux B.C.) at X =0
x
Unsteady state confined aquifer
GW solution
 (r,t)  o  (r,t) 
Q
W (u )
4 T
Sr 2
u 
4T t
u2
u3
W (u )  0 .5 7 7 2  l nu  u 

2 x 2!
3 x 3!
When u is smaller than 0.01, then,
W (u )  0 .5772  lnu
In which conditions is the u small?
Q
Sr 2
Q
2 .2 5Tt
 (r,t) 
(0 .5 7 7 2  l n
)
ln( 2
)
4 T
4Tt
4 T
r S
1/ 2
R
Tt 
 1 .5 
S 



Radius of Influence (u < 0.01)
Unsteady flow to a well
(unconfined aquifer)


 0

  2h
S y h
1 h 


2
(
) 

2
K
t
r r 
 r

 H o h
h
1 
S y ( ) K
t
r r
2
  
2H o
'
 S 'r2
Q

W 
 4Tt
4 T

Corrected drawdown





h

 rh r

Image Well Theory
(2) Recharge Boundary
d
Fully penetrating
stream
Constant
head at X =0
Q
Pumping
Well
x
Image Well Theory
(2) Recharge Boundary: Find drawdown
at the observation well.
Q
d
Image
Well
Q
d
r2
Pumping
Well
r1
Observation Well
Fully penetrating
stream
Constant
head at X =0
x
Image Well Theory
(3) Between Barrier Boundaries
Q
Pumping
Well
Image Well Theory
(4) Barrier-Recharge Boundaries
Q
Pumping
Well
Fully penetrating
stream