Transcript Document

Unconfined Aquifer
Water Table: Subdued replica of the topography
Hal Levin demonstration
after Fetter
http://www.uwsp.edu/water/portage/undrstnd/aquifer.htm
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
Dissolved Solids
mg/l
Cambrian-Ordovician aquifer
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 Wells
in the principal aquifers
of the three principal
groundwater provinces
Alluvial Valleys
& SE Lowland
USGS 1967
Osage &
Till Plains
Springfield
Plateau
<--Maquoketa Shale
Ozark Aquifer
<--Davis/Derby-Doe Run
St. Francois
Aquifer
0
500 gpm
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 = f = Sy + Sr + unconnected porosity
Ss = specific storage
Units: 1/length
= Volume H2O released from storage /unit vol. aquifer /unit head drop
Ss = r g (B + f b)
~ 10-5 /m for sandy gravel
where
(F&C p. 58)
B= aquifer compressibility
b = water compressibility
f = porosity
Storativity S
S
Units: dimensionless
= 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):
D
Freeze & Cherry p. 61
= T/S
Transmissivity T /Storativity S
= K/Ss
Hydraulic Conductivity K/ Specific Storage Ss
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=fi+gj+hk
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
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:

f (x + x, y)  f (x, y)
w
= lim
x0
x
x
f (x, y + y)  f (x, y)
w
= lim
y0
y
y
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”

w ˆ w
w
ˆ
ˆ
w = i
+ j
+ k
x
y
z



ˆ
ˆ
ˆ
 = i
+ j
+ k
x
y
z
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.
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}:
For v(x, y, z) = v1ˆi + v2 ˆj + v3kˆ
Type 1: the curl of v is a vector:
 ˆi
ˆj kˆ 
  
 v 3 v2 
 v1 v 3  ˆ  v2 v1  ˆ
ˆ
Curl v =   v = 


 = 
j + 
i +  
k
 z x 
 y z 
 x y 
x y z 
 v1 v2 v 3 
Type 2: the divergence of v is a scalar:
 


ˆ
ˆ
ˆ
Div v =  v = i
+ j
+ k
y
z
 x
So:

  v1ˆi + v2 ˆj + v3kˆ

(
)
v1
v2
v3
v =
+
+
x
y
z
Great utility for fluxes & material balance
Significance of Divergence

Fy 
dy dx dz
 Fy +

y


Measure of stuff in - stuff out

dz
dy
dx
Fy dx dz

Overall Difference
Rate of Gain in box

 Fx Fy Fz 
+
+

 dx dy dz
 x y z 
rc
T
dx dy dz
t
Laplacian:
2
2
2






2
  = div grad  =    =
+ 2 + 2
2
x
y
z
Gauss Divergence Theorem:
 u dV =  un dA
where un is the surface normal
Continuity Equation (Mass conservation):
rf
=  q + A
t
Steady Flow
q = 0
A = source or sink term;
f = flow porosity
No sources or sinks
Steady, Incompressible Flow

Because the Mass Flux qm :
u = 0
qm = r u
r = constant
Continuity Equation (Mass conservation):
rf
=  q + A
t
A = source or sink term; f = flow porosity
qy
qz
qx
  qm =
+
+
x
y
z
   h 
  h 
  h  
= r  K x  +
K z  
K y  +


 z  

x

x

y

y

z




= r K 2 h
for K x = K y = K z


rf
h
= r Ss
t
t
So, “Diffusion Equation”
where Ss = specific storage
h
K  h = Ss
t
2
“Diffusion Equation”
Cartesian Coordinates

Cylindrical Coordinates
Cylindrical Coordinates,
Radial Symmetry ∂h/∂f = 0
Cylindrical Coordinates,
Purely Radial Flow
∂h/∂f = 0 ∂h/∂z = 0
h
Ss
= K 2h
t
 2h +  2h +  2h = S s h
K t
x2 y2 z2
2
2
1  rh + 1  h +  h = S s h
r r r r2 2 z2
K t
2
Kr

h
Kr 2 + r
r
 2h + 1
r2 r
h
r
h
r
2
h

h

S
+ Kz 2 = s
z
t
Ss h
=K
r t
=
S h
T t
Derivative of Integrals:
v(x )
d
dx
u(x)
f(t) dt = f[v(x)] dv – f[u(x)]du
dx
dx
Thomas p. 539
q 
d q
dq
dp
 f (x, a)dx = 
 f (x, a)dx + f (q, a)  f ( p, a)
da p
da
da
p a
CRC Handbook
“del operator”



ˆ
ˆ
ˆ
 = i
+ j
+ k
x
y
z
Gradient:

w ˆ w
w
ˆ
ˆ
w = i
+ j
+ k
x
y
z

v1
v2
v3
v =
+
+
x
y
z
Divergence:
Diffusion Equation:

h
Ss
= K 2h
t
Darcy's Law:
Hubbert (1940;
k  P 
= g 
 =
n
r 
qv
-
kg
n
J. Geol. 48, p. 785-944)
h
=  Kh
= (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)
n  Kinematic viscosity,
cm2/sec

Gravitational Potential g
GM
g =
r
Gravitational Potential g
GM
g =
r
GM
 g =  2 = Force
r
2 g = 4Gr
If fdx +gdy+hdz is an “exact differential” (= du), then it is easy to integrate,
and the line integral is independent of the path:
Q
Q
P
P
 fdx + gdy + hdz =  du = u(Q)  u(P )
Exact differential:

If true:

u
u
u
du =
dx +
dy +
dz
x
y
z
f=
u
x
g=
u
y
h=
u
z
Condition
 for exactness:
h g
=
y z
=> Curl u = 0
f h
=
z x
g f
=
x y
Conservative Forces
Suppose that force F = fi +gj + hk acts on a line segment dl = idx+jdy+kdz :
Q
Q
P
P
(
)(
)
Q
Work =  F  dl =  fˆi + gˆj + hkˆ  ˆi dx + ˆjdy + kˆdz =  fdx + gdy + hdz
Q
Q
P
P
P
= (if exact) =  u  dr =  du = u(Q)  u(P)

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
 F =  = 0
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
No Flow
Fetter
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
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)
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 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)
Effect of Topography on Regional Groundwater Flow
after Freeze and Witherspoon 1967
http://wlapwww.gov.bc.ca/wat/gws/gwbc/!!gwbc.html
q v =  Kh
r
=  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
Saltwater Intrusion
Saltwater-Freshwater Interface: Sharp gradient in water quality
Seawater
Salinity = 35‰ =
NaCl type water
rsw = 1.025
35,000 ppm =
35 g/l
Freshwater
< 500 ppm (MCL), mostly
Chemically variable; commonly Na Ca HCO3 water
rfw = 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
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
Union Beach, NJ
Water Level & Chlorinity
Craig et al 1996
Fresh Water-Salt Water
Interface?
Air

Fresh Water
hf
r=1.00
?
Salt Water
r=1.025
? ?
Sea level
Ghyben-Herzberg

hf
Sea level
Fresh Water
z
z
Salt Water
P
Ghyben-Herzberg Analysis
Hydrostatic Condition P - rg = 0
Note:
z = depth
rfw = 1.00
No horizontal P gradients
rsw= 1.025
P = gzr sw = g(h f + z)r fw
r fw
z = hf
 40h f
r sw  r fw
Ghyben-Herzberg
r fw
z = hf
 40h f
r sw  r fw

hf
Sea level
Fresh Water
z
z
Salt Water
P
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 out
Water 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
Fresh Water
Equipotentials 
…
.
..
Fresh Water
Equipotentials 
after Ford & Williams 1989
For any two fluids, two head conditions:
Psw = rswg (hsw + z)
and
Pfw = rfw g (hfw + z)
On the mutual interface, Psw = Pfw so:
r fw h fw  r sw hsw
z=
r sw  r fw
Take ∂/∂z and ∂/∂x on the interface,
noting that hsw is a constant as SW is not in motion
r fw h fw 
1=
r sw  r fwz
r fw
h fw
z
=
x r sw  r fw x
∂z/∂x gives slope of interface ~ 40x slope of water table
Also, 40 = spacing of horizontal FW equipotentials in the SW region
Fresh Water Lens
on Island
Saline ground water 0
0
0
Saline ground water 0
after USGS WSP 2250
Confined
Unconfined
Fetter
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 >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 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; D&S p. 79)
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 poolascending 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 a
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 screencan 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.
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