Transcript Document

Darcy Lab: Describe Apparatus
Q
cm3/sec
=
K A ∂h/∂x
= cm/sec cm2
cm/cm
Flow toward Pumping Well,
next to river = line source
= constant head boundary
Plan view
River Channel
Line Source
after Domenico & Schwartz (1990)
Flow Nets:
Set of intersecting Equipotential lines and Flowlines
Flowlines = Streamlines
= Instantaneous flow directions
Pathlines
= Actual particle path
Pathlines ≠ Flowlines for transient flow
Flowlines | to Equipotential surface if K is isotropic
Can be conceptualized in 3D
Flow Net Rules:
Flowlines are perpendicular to equipotential lines (isotropic case)
Spacing between equipotential lines L:
If spacing between lines is constant, then K is constant
In general
K1 m1/L1 = K2 m2/L2
where m = x-sect thickness of aquifer;
L = distance between equipotential lines
For layer of const thickness,
K1/L1 ~ K2/L2
No Flow Boundaries
Equipotential lines meet No Flow boundaries at right angles
Flowlines are tangent to such boundaries (// flow)
Constant Head Boundaries
Equipotential lines are parallel to constant head boundaries
Flow is perpendicular to constant head boundary
FLOW NETS
Impermeble
Boundary
Constant Head
Boundary
Water Table
Boundary
after Freeze & Cherry
MK Hubbert
1903-1989
http://photos.aip.org/
MK Hubbert (1940)
http://www.wda-consultants.com/java_frame.htm?page17
Consider piezometers emplaced near hilltop & near valley
MK Hubbert (1940)
http://www.wda-consultants.com/java_frame.htm?page17
Fetter, after Hubbert (1940)
Fetter, after Hubbert (1940)
Cedar Bog, OH
Piezometer
Cedar Bog, Ohio
Topographic Highs tend to be Recharge Zones
h decreases with depth
Water tends to move downward => recharge zone
Topographic Lows tend to be Discharge Zones
h increases with depth
Water will tend to move upward => discharge zone
It is possible to have flowing well in such areas,
if case the well to depth where h > h@ sfc.
Hinge Line:
Separates recharge (downward flow) & discharge areas (upward flow).
Can separate zones of soil moisture deficiency & surplus (e.g., waterlogging).
Topographic Divides constitute Drainage Basin Divides for Surface water
e.g., continental divide
Topographic Divides may or may not be GW Divides
Bluegrass Spring
Criss
MK Hubbert (1940)
http://www.wda-consultants.com/java_frame.htm?page17
Equipotential Lines
Lines of constant head.
Contours on potentiometric surface or on water table map
=> Equipotential Surface in 3D
Potentiometric Surface: ("Piezometric sfc")
Map of the hydraulic head;
Contours are equipotential lines
Imaginary surface representing the level to which water would
rise in a nonpumping well cased to an aquifer,
representing vertical projection of equipotential surface to land sfc.
Vertical planes assumed; no vertical flow:
2D representation of a 3D phenomenon
Concept rigorously valid only for horizontal flow w/i horizontal aquifer
Measure w/ Piezometers = small dia non-pumping well with short screencan measure hydraulic head at a point (Fetter, p. 134)
How do we know basic flownet picture is correct?
How do we know basic flownet picture is correct?
Mathematical solutions (Toth, 1962, 1963)
Numerical Simulations
Data
Basin Geometry: Sinusoidal water table on a regional topo slope
Toth (1962, 1963)
h(x, z0) =
z0
+
constant +
Bx/L
+ b sin (2px/l)
regional slope +
local relief
Sinusoidal Water Table with a Regional Slope
Z
Z= Z
B
0
X= X
0
Distance, x
X= L
Basin Geometry: Sinusoidal water table on a regional topo slope
Toth (1962, 1963)
h(x, z0) =
z0
+
Bx/L
+ b sin (2px/l)
constant + regional slope + local relief
Solve Laplace’s equation
h = 0
2
Simulate nested set of flow systems
x, z = A' B'

How do we get q?
Coshp z / L Cosp x / L
Coshp z0 / L
e.g., D&S
Discharge
Recharge
No Flow
Regional flow pattern in an area of sloping topography and water table.
Fetter, after Toth (1962) JGR 67, 4375-87.
Systems
Local
Flow
Intermediate
Flow System
Regional
Flow System
Australian Government
after Toth 1963
Conclusions
General slope causes regional GW flow system,
If too small, get only local systems
If the regional slope and relief are both significant, get regional,
intermediate, and local GW flow systems.
Local relief causes local systems.
The greater the amplitude of the relief, the greater the proportion of the water in the local system
If the regional slope and relief are both negligible, get flat water table
often with waterlogged areas mostly discharged by ET
For a given water table, the deeper the basin, the more important the
regional flow
High relief & deep basins promote deep circulation into hi T zones
End 24
Begin 25
MK Hubbert
1903-1989
FLOW NETS
Flow Line
AIP
Equipotential Line
Hubbert (1940)
http://www.wda-consultants.com/java_frame.htm?page17
How do we know basic flownet picture is correct?
Data
Mathematical solutions (Toth, 1962, 1963)
Numerical Simulations
Piezometer
Cedar Bog, Ohio
Pierre Simon
Laplace
1749-1827

h = 0
2
Discharge
Recharge
No Flow
x, z = A' B'
Coshp z / L Cosp x / L
Coshp z0 / L
Regional flow pattern in an area of sloping topography and water table.
Fetter, after Toth (1962) JGR 67, 4375-87.
Numerical Simulations
Basically reproduce Toth’s patterns
High K layers act as “pirating agents
Refraction of flow lines tends to align flow parallel to hi K layer,
and perpendicular to low K layers
Isotropic Systems
Regular slope
Sinusoidal slope
Effect of Topography on Regional Groundwater Flow
after Freeze and Witherspoon 1967
http://wlapwww.gov.bc.ca/wat/gws/gwbc/!!gwbc.html
Isotropic Aquifer
Anisotropic Aquifer
Kx: Kz = 10:1
after Freeze *& Witherspoon 1967
Layered Aquifers
after Freeze *& Witherspoon 1967
Confined Aquifers
Sloping Confining Layer
Horizontal Confining Layer
after Freeze *& Witherspoon 1967
Conclusions
General slope causes regional GW flow system,
If too small, get only local systems
Local relief causes local systems.
The greater the amplitude of the relief, the greater the proportion of the water in the local system
If the regional slope and relief are both negligible, get flat water table
often with waterlogged areas mostly discharged by ET
If the regional slope and relief are both significant, get regional,
intermediate, and local GW flow systems.
For a given water table, the deeper the basin, the more important the
regional flow
High relief & deep basins promote deep circulation into hi T zones
Flow in a Horizontal Layers
Case 1: Steady Flow in a Horizontal Confined Aquifer
Darcy Velocity q:
Flow/ unit width:

Q
q =
=  K h
A
Q' =  K m h
Typically have equally-spaced equipotential lines

Case 2: Steady Flow in a Horizontal, Unconfined Aquifer
Dupuit (1863) Assumptions:
Grad h = slope of the water table
Equipotential lines (planes) are vertical
Streamlines are horizontal
Flow/ unit width:
m2/s
K
2
Q' =  K h h = 
h
2
Q’dx = -K h dh
 K  h dh
0
h1
 Q'dx =

Q' L = 


h2
L
K 2 2
h2 - h1 

2
2
2

K h2 - h1
Q' =  

2  L 
Dupuit
Equation
Fetter p. 164
Dupuit eq.
Head h, m
20
 2 2Q' x 0.5
h =  h1 


K 

15


10
h
K = 10
5
-5
m/s
Q' = 8 x10
-5
2
m /s
0
0
5
10
15
20
Impervious
Base
Distance x, m
Steady flow
No sources or sinks
cf. Fetter p. 164
Better Approach
Q’ = -K h dh/dx
dQ’/dx = 0 continuity equation
So:
d2h2
= 0
2
dx
for one dimensional flow
More generally, for an Unconfined Aquifer:
Steady
 flow:
No sources or sinks
Laplace’s equation in h2
Steady flow
with source term:
Poisson Eq in 
h2
d2h2 d2h2
2 2
 2 = 0 = h
2
dx
dy
K 2 2
- h = w
2
where w = recharge cm/sec
cf. Fetter p. 167
F&C 189
Steady unconfined flow:
with a source term
Poisson Eq in h2
K 2 2
- h = w
2
1-D
K  h
2
2 x
2

Solution:

2
= w
2
wx
h2 = + Ax + B
K
Boundary conditions: @ x= 0 h= h1 ; @ x= L h= h2
 h 2 = wL  xx - h  h x + h 2
1
K
L
2
1
2
2
cf. Fetter p. 167
F&C 189
Unconfined flow with recharge
30
w = 10-8 m/s
K = 10-5 m/s
@ x=0 h1 = 20m
Head h, m
25
@ x=1000m h2 = 10m
20
w
15
10
2
2
h

h
x
w
L

x
x




1
2
2
h =
+ h12
K
L
5

-200
0
200
400
600
800
1000
1200
Distance x, m
cf. Fetter p. 167
F&C 189

Finally, for unsteady unconfined flow: Boussinesq Eq.
  h 
  h  S y h
h  +
h  =
x  x 
y  y  T t
Sy is specific yield
Fetter p. 150-1
For small drawdown compared to saturated thickness b:
Linearized Boussinesq Eq. (Bear p. 408-9)
S y h
h
h
+
=
2
2
x
y
Kb t
2
h = 0
2
Laplace’s Equation
Steady flow
2
A
h =
T
2
Poisson’s Equation
Steady Flow with Source or Sink
S h
h =
T t
2
Diffusion Equation
End Part II
Pierre Simon Laplace
1749-1827
Dibner Lib.
MK Hubbert
1903-1989
http://upload.wikimedia.org/wikipedia/en/f/f7/Hubbert.jpg
Leonhard Euler
1707 - 1783
wikimedia.org
Charles V. Theis
19-19
http://photos.aip.org/
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
After Toth 1983
after Johnson 1975
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 
2pT
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 
2pT
h(, t ) = h0

Solution:
“Theis equation”
or “Non-equilibrium Eq.”
where
Q
Drawdown = h0  h =
W (u)
4 pT

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 pT  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
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
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
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)
http://www.co.portage.wi.us/Groundwater/undrstnd/topo.htm
after Toth 1963
Australian Government
after Toth 1963
PROBLEMS OF GROUNDWATER USE
Saltwater Intrusion
Mostly a problem in coastal areas: GA NY FL Los Angeles
Abandonment of freshwater wells; e.g., Union Beach, NJ
Los Angeles & Orange Ventura Co; Salinas & Pajaro Valleys; Fremont
Water level have dropped as much as 200' since 1950.
Correct with artificial recharge
Upconing of underlying brines in Central Valley
Saltwater Intrusion
Saltwater-Freshwater Interface: Sharp gradient in water quality
Seawater
Salinity = 35‰ =
NaCl type water
sw = 1.025
35,000 ppm =
35 g/l
Freshwater
< 500 ppm (MCL), mostly
Chemically variable; commonly Na Ca HCO3 water
fw = 1.000
Nonlinear Mixing Effect:
Dissolution of cc @ mixing zone of fw & sw
Possible example: Lower Floridan Aquifer: mostly 1500’ thick
Very Hi T ~ 107 ft2/day in “Boulder Zone” near base, f~30% paleokarst?
Cave spongework
Clarence King
1st Director of USGS
1879-1881