Darcy Lab: Describe Apparatus

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Darcy Lab Describe Apparatus Q K
A ?h/?x cm3/sec cm/sec cm2
cm/cm
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Flow toward Pumping Well, next to river line
source constant head
boundary Plan view
after Domenico Schwartz (1990)
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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
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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
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FLOW NETS
Impermeble Boundary
Constant Head Boundary
Water Table Boundary
after Freeze Cherry
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MK Hubbert 1903-1989
http//photos.aip.org/
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MK Hubbert (1940) http//www.wda-consultants.com/j
ava_frame.htm?page17
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Consider piezometers emplaced near hilltop near
valley
MK Hubbert (1940) http//www.wda-consultants.com/j
ava_frame.htm?page17
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Fetter, after Hubbert (1940)
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Fetter, after Hubbert (1940)
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Cedar Bog, OH
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Piezometer Cedar Bog, Ohio
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Topographic Highs tend to be Recharge Zones h
decreases with depth Water tends to move
downward gt recharge zone Topographic Lows
tend to be Discharge Zones h increases with
depth Water will tend to move upward gt
discharge zone It is possible to have flowing
well in such areas, if case the well to depth
where h gt h_at_ 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
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Bluegrass Spring
Criss
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MK Hubbert (1940) http//www.wda-consultants.com/j
ava_frame.htm?page17
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Equipotential Lines Lines of constant head.
Contours on potentiometric surface or on water
table map gt 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
screen- can measure hydraulic head at a point
(Fetter, p. 134)
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How do we know basic flownet picture is correct?
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How do we know basic flownet picture is
correct? Mathematical solutions (Toth, 1962,
1963) Numerical Simulations Data
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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
B
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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 Laplaces equation Simulate nested set of
flow systems
e.g., DS
How do we get q?
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Regional flow pattern in an area of sloping
topography and water table. Fetter, after
Toth (1962) JGR 67, 4375-87.
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after Toth 1963
Australian Government
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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
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End 24 Begin 25
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MK Hubbert 1903-1989
AIP
Hubbert (1940)
http//www.wda-consultants.com/java_frame.htm?page
17
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How do we know basic flownet picture is
correct? Data Mathematical solutions (Toth,
1962, 1963) Numerical Simulations
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Piezometer Cedar Bog, Ohio
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Pierre Simon Laplace 1749-1827
Regional flow pattern in an area of sloping
topography and water table. Fetter, after
Toth (1962) JGR 67, 4375-87.
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Numerical Simulations Basically reproduce
Toths 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
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Isotropic Systems
Regular slope
Sinusoidal slope
Effect of Topography on Regional Groundwater Flow
after Freeze and Witherspoon 1967 http//wlapwww.g
ov.bc.ca/wat/gws/gwbc/!!gwbc.html
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Isotropic Aquifer
Anisotropic Aquifer Kx Kz 101
after Freeze Witherspoon 1967
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Layered Aquifers
after Freeze Witherspoon 1967
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Confined Aquifers
Sloping Confining Layer
Horizontal Confining Layer
after Freeze Witherspoon 1967
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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
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Flow in a Horizontal Layers Case 1 Steady Flow
in a Horizontal Confined Aquifer
Darcy Velocity q
Flow/ unit width
Typically have equally-spaced equipotential lines
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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
Qdx -K h dh
Dupuit Equation Fetter p. 164
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Steady flow No sources or sinks
cf. Fetter p. 164
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Better Approach
Q -K h dh/dx
dQ/dx 0 continuity equation
So
for one dimensional flow
More generally, for an Unconfined Aquifer
Steady flow No sources or sinks Laplaces
equation in h2
Steady flow with source term Poisson Eq in h2
where w recharge cm/sec
cf. Fetter p. 167 FC 189
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Steady unconfined flow with a source
term Poisson Eq in h2
1-D
Solution
Boundary conditions _at_ x 0 h h1 _at_ x
L h h2
cf. Fetter p. 167 FC 189
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Unconfined flow with recharge
w
cf. Fetter p. 167 FC 189
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Finally, for unsteady unconfined flow Boussinesq
Eq.
Sy is specific yield Fetter p.
150-1
For small drawdown compared to saturated
thickness b Linearized Boussinesq Eq. (Bear
p. 408-9)
Laplaces Equation Steady flow
Diffusion Equation
Poissons Equation Steady Flow with Source or Sink
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End Part II
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Pierre Simon Laplace 1749-1827
Dibner Lib.
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MK Hubbert 1903-1989
http//upload.wikimedia.org/wikipedia/en/f/f7/Hubb
ert.jpg
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Leonhard Euler 1707 - 1783
wikimedia.org
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Charles V. Theis 19-19
http//photos.aip.org/
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for unconfined flow
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After Toth 1983
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after Johnson 1975
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Radial flow
Transient flow, Confined Aquifer, No
recharge Constant pumping rate Q
Initial Condition Boundary conditions
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Radial flow
Initial Condition Boundary conditions
Solution Theis equation or
Non-equilibrium Eq.
where
and where
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Approximation for t gtgt 0
DS p. 151
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Pumping of Confined Aquifer
Not GW level Potentiometric sfc!
USGS Circ 1186
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Pumping of Unconfined Aquifer
USGS Circ 1186
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Santa Cruz River Martinez Hill, South of Tucson
AZ
1942 Cottonwoods, Mesquite
1989 gt100 GW drop
USGS Circ 1186
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for unconfined flow
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Initial Condition
Pumping _at_ rate Q1 (note divide)
Pumping _at_ rate Q2 gtQ1
USGS Circ 1186
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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)
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after Toth 1963
http//www.co.portage.wi.us/Groundwater/undrstnd/t
opo.htm
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after Toth 1963
Australian Government
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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
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Saltwater Intrusion Saltwater-Freshwater
Interface Sharp gradient in water quality
Seawater Salinity 35 35,000
ppm 35 g/l NaCl type water
rsw 1.025 Freshwater lt 500
ppm (MCL), mostly Chemically variable
commonly Na Ca HCO3 water rfw
1.000 Nonlinear Mixing Effect Dissolution of
cc _at_ mixing zone of fw sw Possible
example Lower Floridan Aquifer mostly 1500
thick Very Hi T 107 ft2/day in Boulder
Zone near base, f30 paleokarst? Cave
spongework
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Clarence King 1st Director of USGS 1879-1881
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