Upload
silvia-mcdowell
View
221
Download
3
Embed Size (px)
Citation preview
after Fetterhttp://www.uwsp.edu/water/portage/undrstnd/aquifer.htm
Unconfined AquiferWater Table: Subdued replica of the topography Hal Levin demonstration
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)
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!
Hydrostratigraphic Unit: e.g. MO, IL C-Ord sequence of dolostone & sandstone capped by Maquoketa shale
Aquifer Types
Cambrian-Ordovician aquifer
Dissolved Solidsmg/l
http://capp.water.usgs.gov/gwa/ch_d/gif/D112.GIF
USGS
http://capp.water.usgs.gov/gwa/ch_d/gif/D112.GIF
Typical Yields of Wellsin the principal aquifers of the three principal
groundwater provinces USGS 1967
Alluvial Valleys & SE Lowland
Osage & Till Plains
Springfield Plateau
Ozark Aquifer
St. Francois Aquifer
<--Maquoketa Shale
<--Davis/Derby-Doe Run
0 500 gpm
Ss = specific storage Units: 1/length
= Volume H2O released from storage /unit vol. aquifer /unit head drop (F&C p. 58)
Ss = g B where aquifer compressibility ~ 10-5 /m for sandy gravel = water compressibility
= porosity
Sy = Specific yield Units: dimensionless
= storativity for an unconfined aquifer "unconfined storativity"
= Vol of H2O drained from storage/total volume rock (D&S, 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 F&C, p. 61
Specific retention: Sr = = Sy + Sr + unconnected porosity
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 >> mSs
For confined aquifers, typically S = 0.005 to 0.00005
Transmissivity T = K*m 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
HYDRAULIC DIFFUSIVITY (D): Freeze & Cherry p. 61
D = T/S Transmissivity T /Storativity S
= K/Ss Hydraulic Conductivity K/ Specific Storage Ss
FUNDAMENTAL CONCEPTS AND PARTIAL DERIVATIVESScalars: 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| cos commutative
Cross Product (vector product): a x b = - b x a = |a| |b| sin 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
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:
€
∂w∂x
= Δx→0lim
f x + Δx , y( ) − f x , y( )Δx
€
∂w∂y
= Δy→0lim
f x , y + Δy( ) − f x , y( )Δy
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.
€
∇ = ˆ i ∂∂x
+ j ∂∂y
+ ˆ k ∂∂z
€
∇w = ˆ i ∂w∂x
+ j ∂w∂y
+ ˆ k ∂w∂z
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
€
Curl v = ∇ × v =
ˆ i ˆ j ˆ k ∂∂x
∂∂y
∂∂z
v1 v2 v3
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ =
∂v3
∂y−
∂v2
∂z
⎛
⎝ ⎜
⎞
⎠ ⎟ ˆ i +
∂v1
∂z−
∂v3
∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟ j +
∂v2
∂x−
∂v1
∂y
⎛
⎝ ⎜
⎞
⎠ ⎟ ˆ k
€
Div v = ∇ • v = ˆ i ∂∂x
+ j ∂∂y
+ ˆ k ∂∂z
⎛
⎝ ⎜
⎞
⎠ ⎟• v1
i + v2ˆ j + v3
ˆ k ( )
€
∇ • v = ∂v1
∂x +
∂v2
∂y +
∂v3
∂z
€
For v x , y, z( ) = v1 i + v2
ˆ j + v3ˆ k
dxdy
dz€
Fy +∂Fy
∂ydy
⎛
⎝ ⎜
⎞
⎠ ⎟ dx dz
€
Fy dx dz
Overall Difference
€
∂Fx
∂x+
∂Fy
∂y+
∂Fz
∂z
⎛
⎝ ⎜
⎞
⎠ ⎟ dx dy dz
Rate of Gain in box
€
ρ c∂T∂t
dx dy dz
Significance of DivergenceMeasure of stuff in - stuff out
Laplacian:
€
∇2Φ = div grad Φ = ∇ • ∇Φ = ∂2Φ∂x2 +
∂2Φ∂y2 +
∂2Φ∂z2
Gauss Divergence Theorem:
where un is the surface normal
€
∇ • u dV∫∫∫ = undA ∫∫
Continuity Equation (Mass conservation):
A = source or sink term; = flow porosity
No sources or sinks
= constant
€
∂ρφ∂t
= ∇ • q + A
€
∇ • q = 0 Steady Flow
Steady, Incompressible Flow
€
∇ • u = 0
€
qm = ρ u Because the Mass Flux qm :
Continuity Equation (Mass conservation):
A = source or sink term; = flow porosity
€
K ∇2h = Ss ∂h∂t
So, “Diffusion Equation”
€
∇ • qm = ∂qx
∂x +
∂qy
∂y +
∂qz
∂z
€
= ρ ∂∂x
K x
∂h∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟ +
∂∂y
K y
∂h∂y
⎛
⎝ ⎜
⎞
⎠ ⎟ +
∂∂z
K z
∂h∂z
⎛ ⎝ ⎜
⎞ ⎠ ⎟
⎧ ⎨ ⎩
⎫ ⎬ ⎭
€
= ρ K ∇2h for K x = K y = K z
€
∂ρφ∂t
= ρ Ss
∂h∂t
where Ss = specific storage
€
∂ρφ∂t
= ∇ • q + A
∂
2
h
∂ x
2
+
∂
2
h
∂ y
2
+
∂
2
h
∂ z
2
=
Ss
K
∂ h
∂ t
1
r
∂
∂ r
r
∂ h
∂ r
+
1
r
2
∂
2
h
∂ ϕ
2
+
∂
2
h
∂ z
2
=
Ss
K
∂ h
∂ t
Kr
∂
2
h
∂ r
2
+
Kr
r
∂ h
∂ r
+ Kz
∂
2
h
∂ z
2
= Ss
∂ h
∂ t
∂
2
h
∂ r
2
+
1
r
∂ h
∂ r
=
Ss
Kr
∂ h
∂ t
=
S
T
∂ h
∂ t
Cartesian Coordinates
Cylindrical Coordinates
Cylindrical Coordinates,Radial Symmetry ∂h/∂ = 0
Cylindrical Coordinates,Purely Radial Flow ∂h/∂ = 0 ∂h/∂z = 0
€
Ss ∂h∂t
= K ∇2h “Diffusion Equation”
d
dx
f( t )
dt = f[ v ( x )]
dv
dx
– f[ u ( x )]
du
dx
u ( x )
v ( x )
Derivative of Integrals:
€
dda
f x , a( )p
q
∫ dx = ∂∂a
f x , a( )[ ]dx + f q, a( )dqdap
q
∫ − f p, a( )dpda
CRC Handbook
Thomas p. 539
€
Ss ∂h∂t
= K ∇2h €
∇ • v = ∂v1
∂x +
∂v2
∂y +
∂v3
∂z
Gradient:
“del operator”
€
∇ = ˆ i ∂∂x
+ j ∂∂y
+ ˆ k ∂∂z
€
∇w = ˆ i ∂w∂x
+ j ∂w∂y
+ ˆ k ∂w∂z
Divergence:
Diffusion Equation:
Darcy's Law: Hubbert (1940; J. Geol. 48, p. 785-944)
where:
qv Darcy Velocity, Specific Discharge or Fluid volumetric flux vector (cm/sec)
k permeability (cm2)
K = kg/hydraulic conductivity (cm/sec)
Kinematic viscosity, cm2/sec
€
qv = k
νg −
∇P
ρ
⎡
⎣ ⎢
⎤
⎦ ⎥ = -
kg
ν∇h[ ] = − K∇h
= (k/[force/unit mass]
Gravitational Potential g
€
g =GM
r
Gravitational Potential g
€
g =GM
r
∇Φg = −GM
r2= Force
∇2Φg = 4πGρ
€
fdx + gdy + hdz = P
Q∫ du = u(Q )− u(P )
P
Q∫
If fdx +gdy+hdz is an “exact differential” (= du), then it is easy to integrate, and the line integral is independent of the path:
Condition for exactness:
€
∂h∂y
=∂g∂z
∂f∂z
=∂h∂x
∂g∂x
=∂f∂y
€
du = ∂u∂x
dx + ∂u∂y
dy + ∂u∂z
dz Exact differential:
€
f =∂u∂x
g =∂u∂y
h =∂u∂z
If true:
=> Curl u = 0
€
Work = F ⋅dl =P
Q∫ fˆ i + g j + hˆ k ( ) ⋅ ˆ i dx + ˆ j dy + ˆ k dz( ) =
P
Q∫ fdx + gdy + hdz
P
Q∫
Suppose that force F = fi +gj + hk acts on a line segment dl = idx+jdy+kdz :
€
= (if exact) = ∇u ⋅dr =P
Q∫ du = u(Q )− u(P )
P
Q∫
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).
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
In general:
€
∇ × F = ∇ ×∇Φ = 0
Conservative Forces
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
Fetter
No Flow
No
Flow
No Flow
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
Domenico & Schwartz (1990)
Flow beneath DamVertical x-section
Flow toward Pumping Well,next to river = line source
= constant head boundary
Plan view
River Channel
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
MK Hubbert (1940)http://www.wda-consultants.com/java_frame.htm?page17
Fetter, after Hubbert (1940)
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)
after Freeze and Witherspoon 1967http://wlapwww.gov.bc.ca/wat/gws/gwbc/!!gwbc.html
Effect of Topography on Regional Groundwater Flow
€
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
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
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
Craig et al 1996
Union Beach, NJWater Level & Chlorinity
Ghyben-Herzberg
Air
Fresh Water=1.00hf
Fresh Water-Salt Water Interface?
Sea level
Salt Water=1.025
? ? ?
Ghyben-Herzberg
Salt Water
Fresh Water
hf
z
Ghyben-Herzberg
P
Sea level
zinterface
€
P = gzρ sw = g(h f + z)ρ fw
z = h fρ fw
ρ sw −ρ fw
≈ 40h f
Ghyben-Herzberg Analysis
Hydrostatic Condition P - g = 0 No horizontal P gradients
Note: z = depth fw = 1.00 sw= 1.025
Ghyben-Herzberg
Salt Water
Fresh Water
hf
z
Ghyben-Herzberg
P
Sea level
zinterface
€
z = h fρ fw
ρ sw −ρ fw
≈ 40h f
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 1000’s ft below sea level; e.g., Hawaii
Re-entrants in the interface near coastal springs, FLA
Interesting implications:
1) If is 10’ ASL, then interface is 400’ BSL
2) If decreases 5’ ASL, then interface rises 200’ BSL
3) Slope of interface ~ 40 x slope of water table
Hubbert’s (1940) Analysis
Hydrodynamic condition with immiscible fluid interface
1) If hydrostatic conditions existed: All FW would have drained outWater table @ 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
Ford & Williams 1989
….
..
after Ford & Williams 1989
….
..
Fresh Water Equipotentials
Fresh Water Equipotentials
For any two fluids, two head conditions:
Psw = swg (hsw + z) and Pfw = fw g (hfw + z)
On the mutual interface, Psw = Pfw so:
€
1 =ρ fw
ρ sw −ρ fw
∂h fw
∂z
∂z∂x
=ρ fw
ρ sw −ρ fw
∂h fw
∂x
€
€
z =ρ fwh fw −ρ swhsw
ρ sw −ρ fw
∂z/∂x gives slope of interface ~ 40x slope of water table
Also, 40 = spacing of horizontal FW equipotentials in the SW region
Take ∂/∂z and ∂/∂x on the interface, noting that hsw is a constant as SW is not in motion
after USGS WSP 2250
Saline ground water 000
Fresh Water Lenson Island
Saline ground water 0
Confined
Unconfined
Fetter
Saltwater Intrusion
Mostly a problem in coastal areas: GA NY FL Los AngelesFrom above analysis,
if lower by 5’ ASL by pumping, then interface rises 200’ BSL!
Abandonment of freshwater wells- e.g., Union Beach, NJCan attempt to correct with artificial recharge- e.g., Orange CoLos 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 >17 MGD Problems include overpumping;
upconing due to wetlands drainage (Everglades) Marco Island- Hawthorn Fm. @ 540’:
Cl to 4800 mg/l (cf. 250 mg/l Cl drinking water std)
Possible Solutions
Artificial Recharge (most common)
Reduced Pumping
Pumping trough
Artificial pressure ridge
Subsurface Barrier
End
USGS WSP 2250
USGS WSP 2250
USGS WSP 2250
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 constantIn general K1 A1/L1 = K2 A2/L2 where A = x-sect thickness of aquifer;
L = distance between equipotential linesFor layer of const thickness, K1/L1 = K2/L2 (eg. 3.35; D&S p. 79)
FLUID DYNAMICS Consider flow of homogeneous fluid of constant densityFluid 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.
after Toth (1963)http://www.uwsp.edu/water/portage/undrstnd/topo.htm
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 flowConcept 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; D&S 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