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
2h + 2h + 2h = S s h
K t
x2 y2 z2
2
2
1 rh + 1 h + h = S s h
r r r r2 2 z2
K t
2h
Kr
Kr
+ r
r
2
2h + 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 Kh
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 Kh
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
limr
r0 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
limr
r0 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 Kh
= 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