after Fetter

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Unconfined Aquifer Water Table Subdued replica
of the topography
Hal Levin
demonstration
after Fetter http//www.uwsp.edu/water/portage/und
rstnd/aquifer.htm
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Aquifer Types
Unconfined Aquifer aquifer in which the water
table forms upper boundary. Water table
aquifer Head h z P 1 atm e.g.,
Missouri, Mississippi Meramec River valleys
Hi yields, good quality Ogalalla Aquifer
(High Plains aquifer) CO KS NE NM OK SD QT
Sands gravels, alluvial apron off
Rocky Mts.
Perched Aquifer unconfined aquifer above main
water table Generally above a lens of low-k
material. Note- there also is an "inverted"
water table along bottom!
Confined Aquifer aquifer between two aquitards.
Artesian aquifer if the water level in a
well rises above aquifer Flowing Artesian
aquifer if the well level rises above the ground
surface. e.g., Dakota Sandstone east
dipping K sst, from Black Hills- artesian)
Hydrostratigraphic Unit e.g. MO, IL C-Ord
sequence of dolostone sandstone capped by
Maquoketa shale
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Dissolved Solids mg/l
Cambrian-Ordovician aquifer
http//capp.water.usgs.gov/gwa/ch_d/gif/D112.GIF
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USGS
http//capp.water.usgs.gov/gwa/ch_d/gif/D112.GIF
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Typical Yields of Wells in the principal aquifers
of the three principal groundwater provinces
USGS 1967
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Sy Specific yield Units dimensionless
storativity for an unconfined aquifer
"unconfined storativity" Vol of H2O drained
from storage/total volume rock (DS, p. 116)
Vol of H2O released (grav. drained) from
storage/unit area aquifer/unit head drop Sy
Vwd/VT Typically, Sy 0.01 to 0.30
FC, p. 61 Specific retention Sr f
Sy Sr unconnected porosity
Ss specific storage Units 1/length
Volume H2O released from storage /unit vol.
aquifer /unit head drop (FC p. 58) Ss r
g (B f b) where B aquifer
compressibility 10-5 /m for sandy gravel b
water compressibility f porosity
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Storativity S Units dimensionless S
Volume water/unit area/unit head drop
"Storage Coefficient" S m Ss confined
aquifer S Sy m Ss
unconfined note Sy gtgt mSs For confined
aquifers, typically S 0.005 to 0.00005
Transmissivity T Km m aquifer
thickness Units m2/sec Rate of flow
of water thru unit -wide vertical strip of
aquifer under a
unit hyd. Gradient T 0.015 m2/s
in a good aquifer
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HYDRAULIC DIFFUSIVITY (D)
Freeze Cherry p. 61 D T/S
Transmissivity T /Storativity S K/Ss
Hydraulic Conductivity K/ Specific Storage Ss
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FUNDAMENTAL CONCEPTS AND PARTIAL
DERIVATIVES Scalars Indicate scale (e.g.,
mass, Temp, size, ...) Have a
magnitude Vectors Directed line segment,
Have both direction and magnitude e.g.,
velocity, force...) v f i g j h k
where i, j, k are unit vectors
Two types of vector products Dot Product
(scalar product) a. b b. a a b
cosg commutative Cross Product (vector
product) a x b - b x a a b sing
anticommutative
i. i 1 j. j 1 k. k 1 i. j
i. k j. k 0
Scalar Field Assign some magnitude to each point
in space e.g. Temp Vector Field Assign some
vector to each point in space e.g. Velocity
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FUNCTIONS OF TWO OR MORE VARIABLES
Thomas, p. 495 There are many instances in
science and engineering where a quantity is
determined by many parameters. Scalar function
w f(x,y) e.g., Let w be the temperature,
defined at every point in space
Can make a contour map of a scalar function in
the xy plane. Can take the derivative of the
function in any desired direction with
vector calculus ( directional derivative).
Can take the partial derivatives, which tell
how the function varies wrt changes in
only one of its controlling variables.
In x direction, define
In y direction, define
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FUNCTIONS OF TWO OR MORE VARIABLES
Thomas, p. 495 There are many instances in
science and engineering where a quantity is
determined by many parameters. Scalar function
w f(x,y) e.g., Let w be the temperature,
defined at every point in space
Define the Gradient
del operator
The gradient of a scalar function w is a vector
whose direction gives the surface normal and the
direction of maximum change. The magnitude of
the gradient is the maximum value of this
directional derivative. The direction and
magnitude of the gradient are independent of the
particular choice of the coordinate system.
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If the function is a vector (v) rather than a
scalar, there are two different types of
differential operations, somewhat analogous to
the two ways of multiplying two vectors together
i.e. the cross (vector) and dot (scalar)
products
Type 1 the curl of v is a vector
Type 2 the divergence of v is a scalar
So
Great utility for fluxes material balance
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Significance of Divergence Measure of stuff in
- stuff out
Overall Difference
Rate of Gain in box
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Laplacian
Gauss Divergence Theorem
where un is the surface normal
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Continuity Equation (Mass conservation)
A source or sink term f flow porosity
Steady Flow
No sources or sinks
Steady, Incompressible Flow
r constant
Because the Mass Flux qm
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Continuity Equation (Mass conservation)
A source or sink term f flow porosity
where Ss specific storage
So, Diffusion Equation
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Diffusion Equation
Cartesian Coordinates
Cylindrical Coordinates
Cylindrical Coordinates, Radial Symmetry ?h/?f
0
Cylindrical Coordinates, Purely Radial Flow
?h/?f 0 ?h/?z 0
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Derivative of Integrals
Thomas p. 539
CRC Handbook
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del operator
Gradient
Divergence
Diffusion Equation
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Darcy's Law Hubbert (1940
J. Geol. 48, p. 785-944)
(k/n)force/unit mass
where qv ? Darcy Velocity, Specific
Discharge or Fluid volumetric flux
vector (cm/sec) k permeability
(cm2) K kg/n hydraulic conductivity
(cm/sec) ? ???Kinematic viscosity, cm2/sec
?
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Gravitational Potential Fg
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Gravitational Potential Fg
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If fdx gdyhdz is an exact differential (
du), then it is easy to integrate, and the
line integral is independent of the path
Exact differential
If true
Condition for exactness
gt Curl u 0
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Conservative Forces
Suppose that force F fi gj hk acts on a
line segment dl idxjdykdz
If fdx gdy hdz is exact, then the work
integral is independent of the path, and F
represents a conservative force field that is
given by the gradient of a scalar function u
( potential function).
In general
1. Conservative forces are the gradients of some
potential function. 2. The curl of a gradient
field is zero i.e., Curl (grad u) 0
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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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No Flow
No Flow
No Flow
Fetter
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Flow Net Rules No Flow boundaries are
perpendicular to equipotential lines Flowlines
are tangent to such boundaries (//
flow) Constant head boundaries are parallel to
and equal to the equipotential
surface Flow is perpendicular to constant head
boundary
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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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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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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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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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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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for unconfined flow
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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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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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Union Beach, NJ Water Level Chlorinity
Craig et al 1996
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Fresh Water-Salt Water Interface?
Air
?
Fresh Water r1.00
hf
Sea level
? ? ?

Salt Water r1.025
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Ghyben-Herzberg
?
hf
Sea level
Fresh Water
interface
z
z
Salt Water
P
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Ghyben-Herzberg Analysis Hydrostatic Condition
?P - rg 0 No horizontal P gradients Note
z depth rfw 1.00 rsw
1.025
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Ghyben-Herzberg
?
hf
Sea level
Fresh Water
interface
z
z
Salt Water
P
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• Physical Effects
• Tend to have a rather sharp interface, only
  diffuse in detail
• e.g., Halocline in coastal caves
• Get fresh water lens on saline water
• Islands FW to 1000s ft below sea level
  e.g., Hawaii
• Re-entrants in the interface near coastal
  springs, FLA
• Interesting implications
• If ? is 10 ASL, then interface is 400
  BSL
• If ? decreases 5 ASL, then interface rises
  200 BSL
• 3) Slope of interface 40 x slope of water
  table

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• Hubberts (1940) Analysis
• Hydrodynamic condition with immiscible fluid
  interface
• 1) If hydrostatic conditions existed
• All FW would have drained out
• Water table _at_ sea level, everywhere
• w/ SW below
• 2) G-H analysis underestimates the depth to the
  interface
• Assume interface between two immiscible fluids
• Each fluid has its own potential h everywhere,
• even where that fluid is not present!
• FW potentials are horizontal in static SW and air
  zones,
• where heads for latter phases are constant

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.
..
Ford Williams 1989
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Fresh Water Equipotentials ?
.
..
Fresh Water Equipotentials ?
after Ford Williams 1989
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For any two fluids, two head conditions Psw
rswg (hsw z) and Pfw rfw g (hfw
z) On the mutual interface, Psw Pfw so
Take ?/?z and ?/?x on the interface, noting
that hsw is a constant as SW is not in motion
?z/?x gives slope of interface 40x slope of
water table Also, 40 spacing of horizontal FW
equipotentials in the SW region
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Fresh Water Lens on Island
Saline ground water 0
Saline ground water 0
0
0
after USGS WSP 2250
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Confined
Unconfined
Fetter
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Saltwater Intrusion Mostly a problem in
coastal areas GA NY FL Los Angeles From
above analysis, if lower ? by 5 ASL by
pumping, then interface rises 200
BSL! Abandonment of freshwater wells- e.g.,
Union Beach, NJ Can attempt to correct with
artificial recharge- e.g., Orange Co Los
Angeles, Orange, Ventura Counties Salinas
Pajaro Valleys Water level have dropped as
much as 200' since 1950. Correct with
artificial recharge Also, possible upconing of
underlying brines in Central Valley FLA- now
using reverse osmosis to treat saline GW gt17
MGD Problems include overpumping upconing
due to wetlands drainage (Everglades) Marco
Island- Hawthorn Fm. _at_ 540 Cl to 4800
mg/l (cf. 250 mg/l Cl drinking water std)
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Possible Solutions Artificial Recharge (most
common) Reduced Pumping Pumping trough
Artificial pressure ridge Subsurface Barrier
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End
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USGS WSP 2250
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USGS WSP 2250
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USGS WSP 2250
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Potentiometric Surface defines direction of GW
flow Flow at rt angle to equipotential lines
(isotropic case) If spacing between
equipotential lines is const, then K is
constant In general K1 A1/L1 K2 A2/L2
where A x-sect thickness of aquifer L
distance between equipotential lines For layer
of const thickness, K1/L1 K2/L2 (eg. 3.35
DS p. 79)
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FLUID DYNAMICS Consider flow of homogeneous
fluid of constant density Fluid transport in the
Earth's crust is dominated by Viscous,
laminar flow, thru minute cracks and
openings, Slow enough that inertial effects
are negligible. What drives flow within a
porous medium? Down hill? Down
Pressure? Down Head? Consider Case 1
Artesian well- fluid flows uphill. Case 2
Swimming pool- large vertical P gradient, but no
flow. Case3 Convective gyre w/i Swimming
pool- ascending fluid moves from hi to
lo P descending fluid moves from
low to hi P Case 4 Metamorphic rocks and
magmatic systems.
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after Toth (1963) http//www.uwsp.edu/water/portag
e/undrstnd/topo.htm
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Potentiometric Surface ("Piezometric sfc) Map of
the hydraulic head Imaginary surface
representing level to whic water would rise in a
well cased to the aquifer. Vertical planes
assumed no vertical flow Concept rigorously
valid only for horizontal flow w/i horizontal
aquifer Measure w/ Piezometers- small dia well
w. short screen- can measure hydraulic head at
a point (Fetter, p. 134) Potentiometric Surface
defines direction of GW flow Flow at rt angle
to equipotential lines (isotropic case) If
spacing between equipotential lines is const,
then K is constant In general K1/L1 K2/L2
L distance between equipotential lines (eg.
3.35 DS p. 79) For confined aquifers, get
large changes in pressure (head) with virtually
no change in the thickness of the saturated
column. Potentiometric sfc remains above unit
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