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”
Potentiometric sfc!
USGS Circ 1186
Pumping of
Unconfined Aquifer
USGS Circ 1186
End L21: Now, Spring break!
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 u < 0.1
i.e., 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
“Euler’s
constant”
for small u 0.1 ;
i.e., long times or small r
8
6
W(u)
4
2
Well Function W(u)
= - Ei (-u)
0
W(u) ~ -0.577216 - ln(u)
OK f or u < 0.1
-2
0
0.5
1
u
1.5
2
5
"Well Function" W(u)
4
3
W(u)
-0.577216 -ln(u) + u -u2/4
2
-0.577216 - ln(u) + u - u
2
/4
3 - u4/96
4
5
+ u3+/18
u5/600
u /18
-u /96++u
/600
1
0
W(u)
-1
-0.577216 - ln(u)
-2
0
1
2
3
u
4
5
Potentiometric Surface
Q
Pumping
Well
Confined Aquifer
Observation
Well
m
Theis Eq.
for u < 0.1
Q
Drawdown h0 h
W (u)
4 T
2.25 D t
Q
Drawdown h0 h
ln
2
4 T r
Time Drawdown Method:
Use the approximation to calculate T from the head drop in a single observation well
at two times, t1 and t2
2.25 D t1
Q
h0 h1
ln
2
4 T r
2.25 D t 2
Q
h0 h2
ln
2
4 T r
Subtract:
t1
Q
h2 h1
ln
4 T t 2
for u < 0.1
t >> 0 and/or small r
20
Well Function W(u)
Drawdown, m
15
y = 5.7248 + 1.2215x R= 1
Assumed values:
10
Q = 0.1 m3/s
T = 0.015 m2/s
S = 0.0006
D = 25 m2/s
5
Q
2.303 slope
4 T
0
so
-5
-10
-5
0
T = 1295 m2/d = 0.015 m2/s
5
Log T, days
10
15
Potentiometric Surface
Q
Pumping
Well
Confined Aquifer
Observation
Well #1
m
Observation
Well #2
Theis Eq.
for t >> 0
Q
Drawdown h0 h
W (u)
4 T
2.25 D t
Q
Drawdown h0 h
ln
2
4 T r
Distance Drawdown Method:
Use above to calculate T from head the difference between two different observation wells
located at r1 and r2
2.25 D t
Q
h0 h1
ln
2
4 T r1
Subtract:
2.25 D t
Q
h0 h2
ln
2
4 T r2
r2
Q
h2 h1
ln
2 T r1
for t >> 0
Steady state approximated
=> “Theim eq”
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
Steady state ? ! !
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
Steady state ? ! !
Steady Shape
Image Wells:
No Flow
Can duplicate Impermeable Boundary with Pumping Image Well
PUMPING
WELL
PUMPING
IMAGE WELL
PUMPING
WELL
WHERE?
Image Wells:
No Flow
Can duplicate Impermeable Boundary with Pumping Image Well
PUMPING
WELL
PUMPING
IMAGE WELL
PUMPING
WELL
Image
Well
Pumping
Well
t=1
Drawdown
0
t=10
-5
-10
t=100
-15
-20
-4
-2
0
2
4
Radius, r
Drawdown
Q
W u p W ui
4 T
Pumping Pumping
Well
Well
t=1
Impermeable Boundary
No Flow
-5
-10
-15
0
t=10
-5
-10
t=100
-15
DRAWDOWN
Drawdown
0
-20
-20
-4
-2
0
2
4
Radius, r
Drawdown
Q
W u p W ui
4 T
Initial Condition
Pumping
@ rate Q1
(note divide)
Pumping
@ rate Q2 >Q1
USGS Circ 1186
Image Wells:
Constant Head
Can duplicate Constant Head Boundary with Recharging Image Well
PUMPING
WELL
RECHARGING
IMAGE WELL
PUMPING
WELL
Image Wells:
Constant Head
Can duplicate Constant Head Boundary with Recharging Image Well
PUMPING
WELL
RECHARGING
IMAGE WELL
PUMPING
WELL
t = 1, 10, 100
Drawdown
Q
W u p W ui
4 T
Recharge Well (image)
Pumping Well
Flow toward Pumping Well,
next to river = line source
= constant head boundary
Plan view
River Channel
Line Source
after Domenico & Schwartz (1990)
PROBLEMS OF GROUNDWATER USE
Overdraft
Water table decline
Pumpage sometimes many times recharge = Water Mining
Phoenix AZ avg 8' drop per year; total drop = 400' (130m)
Several CA basins avg 2.5' drop per year;
Drop to 6' drop/yr esp. San Joaquin Valley
Ogallala aquifer (= High Plains aquifer)
Chicago once had artesian wells
Water Table Decline or
Artesian Pressure Loss
>12 m
Craig et al
Head Decline
Craig et al. 1996, after WSP 2275
Cones of depression, Sacramento Valley
Criss & Davisson 1996,
after DWR 1986
89°
43°
42°
41°
88°
87°
Milwaukee
Head Decline (1864-1985)
Cambrian-OrdovicianAquifer,
Chicago-Milwaukee Area
Chicago
Young et al. (1988)
AWRA Mon. 13, p. 55
Santa Cruz River
near Tucson AZ
1942
USGS Circ 1186
Santa Cruz River
near Tucson AZ
1989 >100’ GW drop
USGS Circ 1186
End
#23
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
FLOW NETS
Impermeble
Boundary
Constant Head
Boundary
Water Table
Boundary
after Freeze & Cherry
Pumping
Well
Boundary
Impermeable
No Flow
Drawdown
t=1
t =1
0
t =10
-5
-10
t =100
-15
t=10
t=100
-20
-4
-2
0
2
4
Radius, r
Drawdown
Q
W u p W ui
4 T
Santa Cruz River
Martinez Hill,
South of Tucson AZ
1942
Cottonwoods,
Mesquite
1989
>100’ GW drop
USGS Circ 1186
Theis Eq.
for t >> 0
Q
Drawdown h0 h
W (u)
4 T
2.25 D t
Q
Drawdown h0 h
ln
2
4 T r
Time Drawdown Method:
Use approx. to calculate T from the head drop in the observation well at two times, t1 and t2
t1
Q
h2 h1
ln
4 T t 2
for t >> 0
Distance Drawdown Method:
Use above to calculate T from head difference between two observation wells located at r1 and r2
r2
Q
h2 h1
ln
2 T r1
for t >> 0
Steady state approximated
=> “Theim eq”