Transcript Document

“del operator”
Gradient:

Divergence:

Laplacian:

Diffusion Equation:




ˆ
ˆ
ˆ
  i
 j
 k
x
y
z
w ˆ w
w
ˆ
ˆ
w  i
 j
 k
x
y
z
v1
v2
v3
v 


x
y
z
2
2
2

f

f

f
2
 f 


2
2
x
y
z 2
h
Ss
 K 2h
t
“del operator”
Gradient:

Divergence:

Laplacian:

Diffusion Equation:




ˆ
ˆ
ˆ
  i
 j
 k
x
y
z
w ˆ w
w
ˆ
ˆ
w  i
 j
 k
x
y
z
v1
v2
v3
v 


x
y
z
2
2
2

f

f

f
2
 f 


2
2
x
y
z 2
h
Ss
 K 2h
t
“Diffusion Equation”
Cartesian Coordinates

Cylindrical Coordinates
Cylindrical Coordinates,
Radial Symmetry ∂h/∂f= 0
Cylindrical Coordinates,
Purely Radial Flow
∂h/∂f= 0 ∂h/∂z = 0
h
Ss
 K 2h
t
 2h +  2h +  2h = S s h
K t
x2 y2 z2
2
2
1   rh + 1  h +  h = S s h
r r r r2 2 z2
K t
  2h
Kr
Kr
+ r
r
2
 2h + 1
r2 r
h
r
h
r
2
h

h

S
+ Kz 2 = s
z
t
Ss h
=K
r t
=
S h
T t
Flow beneath Dam
Vertical x-section
Flow toward Pumping Well,
next to river = line source
= constant head boundary
Plan view
River Channel
Domenico & Schwartz (1990)
Well Drawdown
Case 1: Confined aquifier, Constant pumping rate Q, Steady radial flow
qv  Kh
Q / A  K
h
for radial flow
r
Q
h
 K
2 r m
r
where m=aquifer thickness
h
Q r dr

   dh
2 mK r r
h
0
0
Q
r0
ln
 h0  h
2 T r
= Drawdown
“Equilibrium” eq.,
or
“Theim” equation
e.g., Fetter eq 7-38
D&S eq. 5.21 2x
1005
Confined Aquifer
Steady Pumping Rate Q
Q/2 T = 5
1000
Head, h
995
990
985
-1000
-500
0
Radius, r
500
1000
Potentiometric Surface
Q
Confined Aquifer
m
Well Drawdown
Case 2: Unconfined aquifier, Constant pumping rate Q, Steady radial flow
qv  Kh
h
Q / A  K
for radial flow
r
Q
h
 K
2 r h
r
where m=aquifer thickness
h
Q r dr

   h dh
2 K r r
h
0
0
Q
r0
ln
 h02  h 2
K r
≠ (Drawdown)2
Unconfined “Theim”
equation
e.g., Fetter eq 7-39
D&S eq. 5.24
Swindle:
1) Singulatity at r = 0
2) Steady state flux impossible without source this problem requires annular source term
3) Purely radial flow impossible for unconfined case
Radial flow
 h
1 h
1 h


2
r
r r
D t
2
Transient flow, Confined Aquifer, No recharge
Constant pumping rate Q

Initial Condition & Boundary conditions:
h(r, 0)  h0
h(, t )  h0
 h  Q
limr 
r0  r  2 T
for t > 0

Radial flow
 h
1 h
1 h


2
r
r r
D t
2
Initial Condition & Boundary conditions:
h(r, 0)  h0
 h  Q
limr 
r0  r  2 T
h(, t )  h0

Solution:
“Theis equation”
or “Non-equilibrium Eq.”
where
Q
Drawdown  h0  h 
W (u)
4 T

e 
u

W (u)  Ei(u)  

and where
for t > 0
W   0
d
W 0  
r 2S
r2
where u =

4tT 4Dt
Approximation for t >> 0
2.25 D t 
Q
Drawdown  h0  h 
ln

2

4 T  r
D&S p. 151
u2
u3
u4
u5
W (u )  Ei(u )  0.577216  ln u  u  


 ....
4 3 3! 4  4! 5  5!
W(u)  0.577216 lnu
for small u  0.1 ;
i.e., long times or small r
Pumping of
Confined Aquifer
Not GW “level”
Potetiometric sfc!
USGS Circ 1186
Pumping of
Unconfined Aquifer
USGS Circ 1186
End L21: Now, Spring break!
FLOW NETS
Impermeble
Boundary
Constant Head
Boundary
Water Table
Boundary
after Freeze & Cherry
Santa Cruz River
near Tucson AZ
1942
USGS Circ 1186
Santa Cruz River
near Tucson AZ
1989 >100’ GW drop
USGS Circ 1186
Santa Cruz River
Martinez Hill,
South of Tucson AZ
1942
Cottonwoods,
Mesquite
1989
>100’ GW drop
USGS Circ 1186
q v   Kh

=  q m + A
t
 q m  0
 u  0
h K 2
=
 h
t Ss
Darcy's Law
Continuity Equation
Steady flow, no sources or sinks
Steady, incompressible flow
K T
Diffusion Eq.,where
= D
Ss S
S y h
  h    h 
=
h  + h 
K t
x  x  y  y 
Boussinesq Eq.
for unconfined flow
Initial Condition
Pumping
@ rate Q1
(note divide)
Pumping
@ rate Q2 >Q1
USGS Circ 1186