Groundwater Pollution
Remediation (NOTE 3)
Joonhong Park
Yonsei CEE Department
2015. 10. 12.
CEE3330 Y2013 WEEK3
Storage Coefficient, S
• Definition: the volume of water released
from storage (ΔVw) per unit decline in
piezometric head (ΔФ) per unit area (A)
of the aquifer
V w
S 
 S sB  S y
A 
HERE: Ss (Specific Storage), Sy (Specific Yield),
B(thickness of aquifer)
• 0<S<Effective Porosity, neff
Specific Storage, Ss (Freeze and Cherry, 1979)
Definition: the volume of water that an aquifer releases
from storage, per volume,
per unit decline in piezometric head
Ss
1 V w
1 V w p
1 V w



w
V a 
V a p  V a p
Here △p (decline in pressure), γw (specific weight of water)
S s   w  p  n w

dV t



Here βp (compressibility of aquifer material) p
d e
dV w
w  
βw (compressibility of water)
dp
1
Vt
1
Vw
Specific Yield (unconfined aquifer) Sy
Sy
V w

A h
Volume of water drained per unit are per unit decline
in piezometric head (water table)
Sy + Sr = n (Sy < n)
Here: Sr is specific retention
n is porosity
Sy: 0.1~0.3
(ex. Clay 0.02, Sandy clay 0.07, Silt 0.18
Fine sand 0.21, Coarse sand 0.27, Dune sand 0.38
Peat 0.44)
Storage Coefficients (S)
in Typical Aquifers
• Confined Aquifer:
V w
S 
 S sB
A 
• Unconfined Aquifer:
V w
S 
Sy
A 
Dupuit Approximation
• Flow is essentially horizontal or may be
treated as such
• Assumption fails in regions where flow has a
large vertical component
• Good approximation if aquifer thickness
varies only gradually
• Applicable to leaky aquifer if leaking is not so
large
Transmissivity
• Rate of flow per unit width through
entire aquifer thickness per unit
piezometric head gradient
T KB
Two-Dimensional GW Flow Equations
Assume horizontal flow (Dupuit approximation) => vertical equipotentials
Confined Aquifer

1 
T     q Zb  q Zt
S

t
r r
Unconfined Aquifer
Sy
h
   K h     h   N  q Zb
t
N: recharge rate
Flow to a well in a confined aquifer
Q
Assumptions:
Ф(r)
qzt
r
qzb

S(
) T
t
  2
1 
1  2 
 2 
  q Zb  q Zt

2
2
 r

r

r
r

r


After the assumptions are considered, the equations can
be simplified as the following equation.
  2
1  
 0
T  2 

r

r

r


1    
 r
  0
T
r r  r 
  
 r
  C 1
 r 
1
  C 1r
r
  C 1 lnr  C
2
Boundary conditions?

Q  K 2rw B
r
At r=rw
  L
At r=L
Incorporating BCs
C1

Q
r

r
2T
Q r
 
2T r
Q
(L ) 
lnL  C 2
2T
r1
Q
(r1)  (L ) 
ln
2T
L
Thiem Equation (1906)
Q
r2
 r2    r1    (r1)   (r2 ) 
ln
2T
r1
Here σ = Фo – Ф (draw down)
RADIUS OF INFLUENCE: distance beyond
which drawdown is negligible.
Steady flow to a well in an unconfined aquifer
Q
Ф(r)
r
Assumptions:
1  
h 
 rKh
  0

r r 
r 
h
 rKh
C1
r
C1
1
h 2
 K

2
r
r
1
 Kh 2  C 1 lnr  C 2
2
Boundary conditions?
Dupuit-Forchheimer Well Discharge
equation.
h r  hL
2
2
h 22  h12
Q
r

ln
K
rL
Q
r2

ln
K
r1
Confined-Unconfined Comparison
Unsteady state confined aquifer
GW solution
 (r,t)  o  (r,t) 
Q
W (u )
4 T
Sr 2
u 
4Tt
u2
u3
W (u )  0 .5772  lnu  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 .25Tt
 (r,t) 
(0 .5772  ln
)
ln( 2
)
4 T
4Tt
4 T
r S
R
Tt
 1 .5 
S

1/ 2




Radius of Influence (u < 0.01)
Theis Solution
Q
 (r,t)  o  (r,t) 
W (u )
4 T

e X
W (u )  X u
dX  Ei(u )
X
Sr 2
u 
4Tt
u2
u3
W (u )  0 .5772  lnu  u 

2 x 2! 3 x 3!
Papadopulos Solution
(Extensions to anisotropic media)
Q
 (r,t)  o  (r,t) 
W (u xy )
4 T eq
T eq  T XX TYY  T XY
u XY

2
2
2

T
y

T
x
 2T xy xy
S
XX
YY


2
4t 
T eq





Cooper-Jacob Solution
(For a small u)
 (r,t)  o  (r,t) 
Q
W (u )
4 T
Sr 2
u 
4Tt
u2
u3
W (u )  0 .5772  lnu  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 .25Tt
 (r,t) 
(0 .5772  ln
)
ln( 2
)
4 T
4Tt
4 T
r S
R
Tt
 1 .5 
S

1/ 2




Radius of Influence (u < 0.01)
Unsteady flow to a well
(unconfined aquifer)
h
1 
) K
t
r r
S y(
S y h
2
(
)
K
t
 H
o

h 


rh
 0


r 

  2h
1 h 


 r 2  r r 


h
2
  
2H o
'
 S 'r 2
Q

W 
4 T
 4Tt
Corrected drawdown




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
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