Darcy Lab: Describe Apparatus
Q = K A ∂h/∂x
cm3/sec = cm/sec cm2 cm/cm
after Domenico & Schwartz (1990)
Flow toward Pumping Well,next to river = line source
= constant head boundary
Plan view
River C
hannel
Line S
ource
Pathlines ≠ Flowlines for transient flow Flowlines | to Equipotential surface if K is isotropic
Can be conceptualized in 3D
Flow Nets: Set of intersecting Equipotential lines and Flowlines
Flowlines Streamlines Instantaneous flow directions
Pathlines Actual particle path
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 constantIn 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 BoundariesEquipotential lines meet No Flow boundaries at right anglesFlowlines are tangent to such boundaries (// flow)
Constant Head BoundariesEquipotential lines are parallel to constant head boundaries Flow is perpendicular to constant head boundary
Impermeble
Boundary
Constant Head
Boundary
Water Table
Boundary
after Freeze & Cherry
FLOW NETS
http://photos.aip.org/
MK Hubbert1903-1989
MK Hubbert (1940)http://www.wda-consultants.com/java_frame.htm?page17
MK Hubbert (1940)http://www.wda-consultants.com/java_frame.htm?page17
Consider piezometers emplaced near hilltop & near valley
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 LinesLines of constant head. Contours on potentiometric surface or on water tablemap
=> 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 phenomenonConcept rigorously valid only for horizontal flow w/i horizontal aquifer
Measure w/ Piezometers small dia non-pumping well with short screen-can 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 + Bx/L + b sin (2x/)
constant + regional slope + local relief
Sinusoidal Water Table with a Regional Slope
Z
Distance, x
Z = Z0
X = X0
X = L
B
Basin Geometry: Sinusoidal water table on a regional topo slope Toth (1962, 1963)
h(x, z0) = z0 + Bx/L + b sin (2x/)
constant + regional slope + local relief
Solve Laplace’s equation
Simulate nested set of flow systems
€
∇2h = 0
e.g., D&S
€
Φ x , z( ) = A'− B' Cosh π z /L[ ] Cos π x /L[ ]
Cosh π z0 /L[ ]
How do we get q?
Regional flow pattern in an area of sloping topography and water table.Fetter, after Toth (1962) JGR 67, 4375-87.
No Flow
No
Flow
No Flow
Discharge Recharge
after Toth 1963Australian Government
Local
Flow
Systems
Intermediate Flow System
RegionalFlow System
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 24Begin 25
Hubbert (1940)
MK Hubbert1903-1989
http://www.wda-consultants.com/java_frame.htm?page17
Equipotential Line
Flow Line
FLOW NETS
AIP
How do we know basic flownet picture is correct?DataMathematical solutions (Toth, 1962, 1963)Numerical Simulations
Piezometer Cedar Bog, Ohio
Regional flow pattern in an area of sloping topography and water table.Fetter, after Toth (1962) JGR 67, 4375-87.
No Flow
No
Flow
No Flow
Discharge Recharge
Pierre Simon Laplace
1749-1827
€
Φ x , z( ) = A'− B' Cosh π z /L[ ] Cos π x /L[ ]
Cosh π z0 /L[ ]
€
∇2h = 0
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
after Freeze and Witherspoon 1967http://wlapwww.gov.bc.ca/wat/gws/gwbc/!!gwbc.html
Effect of Topography on Regional Groundwater Flow
Isotropic Systems
Regular slope
Sinusoidal slope
Isotropic Aquifer
Anisotropic Aquifer Kx: Kz = 10:1
after Freeze *& Witherspoon 1967
after Freeze *& Witherspoon 1967
Layered Aquifers
after Freeze *& Witherspoon 1967
Confined Aquifers
Sloping Confining Layer
Horizontal Confining Layer
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
€
q = QA
= − K ∇h
€
Q' = − K m ∇h Flow/ unit width:
Darcy Velocity q:
Typically have equally-spaced equipotential lines
Case 2: Steady Flow in a Horizontal, Unconfined Aquifer
€
Q' = − K h ∇h = −K 2
∇h2 Flow/ unit width:m2/s
Dupuit (1863) Assumptions:Grad h = slope of the water tableEquipotential lines (planes) are verticalStreamlines are horizontal
Q’dx = -K h dh
€
Q'dx0
L∫ = − K h dh
h1
h2
∫
€
Q'L = −K2
h22 - h1
2( )
€
Q' = −K2
h2
2 - h12
L
⎛
⎝ ⎜
⎞
⎠ ⎟ Dupuit
Equation Fetter p. 164
0
5
10
15
20
0 5 10 15 20
Dupuit eq.
Distance x, m
K = 10-5 m/s
Q' = 8 x10-5 m2/s
Impervious Base
€
∇
h
€
h = h12 −
2Q' x K
⎛ ⎝ ⎜
⎞ ⎠ ⎟0.5
cf. Fetter p. 164
Steady flowNo sources or sinks
cf. Fetter p. 167 F&C 189
Q’ = -K h dh/dx
dQ’/dx = 0 continuity equation
€
d2h2
dx2 = 0
€
d2h2
dx2 +d2h2
dy2 = 0 = ∇2h2
So:
More generally, for an Unconfined Aquifer:
Steady flowwith source term:Poisson Eq in h2
where w = recharge cm/sec
Steady flow:No sources or sinksLaplace’s equation in h2
Better Approach
€
-K 2
∇2h2 = w
for one dimensional flow
€
-K 2
∇2h2 = w Steady unconfined flow:with a source termPoisson Eq in h2
€
-K 2
∂2h2
∂x2 = w 1-D
Solution:
€
h2 = -wx2
K+ Ax + B
Boundary conditions: @ x= 0 h= h1 ; @ x= L h= h2
€
h2 = w L − x( )x
K-
h12 − h2
2( )x
L+ h1
2 cf. Fetter p. 167 F&C 189
cf. Fetter p. 167 F&C 189
w
Unconfined flow with recharge
5
10
15
20
25
30
-200 0 200 400 600 800 1000 1200
Distance x, m
w = 10-8 m/sK = 10-5 m/s@ x=0 h1 = 20m
@ x=1000m h2 = 10m
€
h2 = w L − x( )x
K-
h12 − h2
2( )x
L+ h1
2
Finally, for unsteady unconfined flow: Boussinesq Eq.
€
∂∂x
h∂h∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟ +
∂∂y
h∂h∂y
⎛
⎝ ⎜
⎞
⎠ ⎟=
Sy
T∂h∂t
Sy is specific yield
Fetter p. 150-1
For small drawdown compared to saturated thickness b:Linearized Boussinesq Eq. (Bear p. 408-9)
€
∂2h∂x2 +
∂2h∂y2 =
Sy
Kb∂h∂t
€
∇2h = AT
€
∇2h = 0
€
∇2h = ST
∂h∂t
Laplace’s EquationSteady flow
Poisson’s EquationSteady Flow with Source or Sink
Diffusion Equation
End Part II
Pierre Simon Laplace1749-1827
Dibner Lib.
http://upload.wikimedia.org/wikipedia/en/f/f7/Hubbert.jpg
MK Hubbert1903-1989
wikimedia.org
Leonhard Euler1707 - 1783
http://photos.aip.org/
Charles V. Theis19-19
€
qv = − K∇h Darcy' s Law
∂ρϕ∂t
= ∇ • qm + A Continuity Equation
∇ • qm = 0 Steady flow, no sources or sinks
∇ • u = 0 Steady, incompressible flow
∂h∂t
=K Ss
∇2h Diffusion Eq., where KSs
=TS
= D
Sy
K∂h∂t
= ∂∂x
h∂h∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟ +
∂∂y
h∂h∂y
⎛
⎝ ⎜
⎞
⎠ ⎟ Boussinesq Eq.
for unconfined flow
After Toth 1983
after Johnson 1975
€
∂ 2h
∂ r 2 + 1
r
∂h
∂r =
1
D
∂h
∂t
Radial flow
Initial Condition & Boundary conditions:
€
h(r , 0 )= h0 h(∞, t )= h0 limr→0
r∂h∂r
⎛ ⎝ ⎜
⎞ ⎠ ⎟=
Q2πT
for t > 0
Transient flow, Confined Aquifer, No rechargeConstant pumping rate Q
€
∂ 2h
∂ r 2 + 1
r
∂h
∂r =
1
D
∂h
∂t
Radial flow
Initial Condition & Boundary conditions:
€
h(r , 0 )= h0 h(∞, t )= h0 limr→0
r∂h∂r
⎛ ⎝ ⎜
⎞ ⎠ ⎟=
Q2πT
for t > 0
€
W (u )= −Ei(−u )=e−ξ
ξu
∞∫ dξ where u =
r2S4tT
=r2
4Dt
and where
Solution: “Theis equation” or “Non-equilibrium Eq.”
€
W ∞( ) = 0 W 0( ) = ∞
€
Drawdown = h0 − h =Q
4πTW (u )
where
Approximation for t >> 0
€
Drawdown = h0 − h ≅Q
4πTln
2.25 D t r2
⎛ ⎝ ⎜
⎞ ⎠ ⎟
D&S p. 151
€
W (u )= −Ei(−u )= −0.577216 − lnu + u −u2
4+
u3
3× 3!−
u4
4 × 4!+
u5
5 × 5!− ....
€
W (u ) ≅ −0.577216 − lnu for small u < 0.1 ; i.e., long times or small r
USGS Circ 1186
Pumping of Confined Aquifer
Not GW “level” Potentiometric sfc!
USGS Circ 1186
Pumping of Unconfined Aquifer
USGS Circ 1186
Santa Cruz RiverMartinez Hill,South of Tucson AZ
1989>100’ GW drop
1942Cottonwoods,Mesquite
€
qv = − K∇h Darcy' s Law
∂ρϕ∂t
= ∇ • qm + A Continuity Equation
∇ • qm = 0 Steady flow, no sources or sinks
∇ • u = 0 Steady, incompressible flow
∂h∂t
=K Ss
∇2h Diffusion Eq., where KSs
=TS
= D
Sy
K∂h∂t
= ∂∂x
h∂h∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟ +
∂∂y
h∂h∂y
⎛
⎝ ⎜
⎞
⎠ ⎟ Boussinesq Eq.
for unconfined flow
USGS Circ 1186
Pumping @ rate Q1
(note divide)
Initial Condition
Pumping @ rate Q2 >Q1
-2
0
2
4
6
8
0 0.5 1 1.5 2
Well Function W(u)= - Ei (-u)
W(u)
u
W(u) ~ -0.577216 - ln(u) OK for u < 0.1
Domenico & Schwartz (1990)
Flow beneath DamVertical x-section
Flow toward Pumping Well,next to river = line source
= constant head boundary
Plan view
River Channel
after Toth 1963http://www.co.portage.wi.us/Groundwater/undrstnd/topo.htm
after Toth 1963Australian Government
PROBLEMS OF GROUNDWATER USE
Saltwater IntrusionMostly a problem in coastal areas: GA NY FL Los AngelesAbandonment of freshwater wells; e.g., Union Beach, NJ
Los Angeles & Orange Ventura Co; Salinas & Pajaro Valleys; FremontWater level have dropped as much as 200' since 1950.
Correct with artificial rechargeUpconing of underlying brines in Central Valley
Saltwater Intrusion
Saltwater-Freshwater Interface: Sharp gradient in water quality
Seawater Salinity = 35‰ = 35,000 ppm = 35 g/l
NaCl type water sw = 1.025
Freshwater
< 500 ppm (MCL), mostly Chemically variable; commonly Na Ca HCO3 waterfw = 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,~30% paleokarst?Cave spongework
Clarence King
1st Director of USGS
1879-1881