FLUID STATICS
No flow
Surfaces of const P and r coincide along gravitational equipotential surfaces
dP
 rg
dz
P  rg(h  z)
for constant r
h = head
= scalar; units of meters
= energy/unit weight (energy of position)
P = 1 atm
surface
P ~ 1.3 atm @10 feet
P ~ 1.6 atm
@20 feet
P ~ 2 atm
@33 feet
 P  0.1 bar/m
0
0
-5
0.2
0.6
DEPT H,
m -10
P, bar
0.4
1.2
-15
0.6
1.8
-20
0.8
2.4
-25
3.0
1
-30
Distance
P = 0.1 bar/m
0
0
-5
0.2
DEPT H,
m -10
P, bar
0.4
-15
0.6
-20
0.8
-25
-30
1
Distance
0
0
-5
0.2
0.6
DEPT H,
m -10
P, bar
h
0.4
1.2
-15
0.6
1.8
-20
0.8
2.4
-25
13.0
-30
Distance
PL > Ph
Ph = 0.1 bar/m
FLUID DYNAMICS in PERMEABLE MEDIA
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 permeable medium?
Down hill?
Down Pressure?
Down Head?
What drives flow through a permeable medium?
Consider:
Case 1: Artesian well
Case 2: Swimming pool
Case 3: Convective gyre
Case 4:
Metamorphic and Magmatic Systems
Humble Texas
Flowing 100 years
Hot, sulfur-rich,
artesian water
http://www.texasescapes.com/
TexasGulfCoastTowns/Humble-Texas.htm
0
0
-5
0.2
0.6
DEPT H,
m -10
P, bar
0.4
1.2
-15
0.6
1.8
-20
2.4
0.8
-25
1
3.0
-30
Distance
P = 0.1 bar/m
0
0

-5
0.2
0.6
DEPT H,
m -10
-15
P, bar


0.4
1.2
0.6
1.8
-20
0.8
2.4
-25
-30

13.0
Distance
P = 0.1bar/m
Criss et al 2000
What drives flow within a porous medium?
RESULTS:
Case 1: Artesian well
Fluid flows uphill.
Case 2: Swimming pool Large vertical P gradient,
but no flow.
Case 3: Convective gyre
Ascending fluid moves from high to low P
Descending fluid moves from low to high P
Case 4:
Metamorphic and Magmatic Systems
Fluid flows both toward heat source, then away,
irrespective of pressure
Darcy's Law
Henry Darcy (1856) Sanitation Engineer
Public water supply for Dijon, France.
Filtered water thru large sand column; attached Hg manometers
Observed relationship bt the volumetric flow rate and the hydraulic gradient
Q  (hu -hl)/L
where
(hu -hl) is the difference in upper & lower manometer readings
L is the spacing length
Q = KA(hu-hl)/L
Rewrite Darcy's Law
Specific Discharge:
q
=
=
=
=

Q/A
-K ∆h/∆L
-K ∂h/∂L
-Ki
q = - K h

where
 "Darcy Velocity"
q Volumetric flux; m3/m2-sec
units of velocity, but is a macroscopic quantity
h hydraulic gradient; dimensionless
 i ∂/∂x + j ∂/∂y + k ∂/∂z
K hydraulic conductivity,
units of velocity (m/sec)
GRADIENT LAWS

= - K h
Darcy’s Law
J
= - D C
Fick’s Law of Diffusion

f
= - K T
Fourier’s Law of Heat Flow
i
= (1/R)V
Ohm’s Law
q

Negative sign: flow is down gradient
Actual microscopic velocity (u)
u = q/f = Darcy Velocity/effective porosity
Clearly, u > q
HYDRAULIC CONDUCTIVITY, K
K

m/s
= krg/m
 kg/n
units of velocity
Proportionality constant in Darcy's Law
Property of both fluid and medium
see D&S, p. 62
HYDRAULIC POTENTIAL (F):
F= gh
energy/unit mass
cf. h = energy/unit weight
= gz + P/rw
Consider incompressible fluid element
@ elevation zi= 0
pressure Pi
Move to new position
Energy difference:
Energy/unit mass
ri
and velocity v = 0
z, P, r,v
lift mass
= mg(z- zi)
F g z
+ accelerate + compress (=  VdP)
+ mv2/2
+ m  V/m) dP
latter term = m (1/r)dP
+ v2/2
+  (1/r)dP
For incompressible fluid (r = const) & slow flow (v2/2 0), zi=0,
Energy/unit mass: F g z + P/r
Force/unit mass = F = g - P/r
Force/unit weight = h = 1 - P/rg
= gh
Pi = 0
Rewrite Darcy's Law:
qm  r qv 


q
m

Hubbert (1940,
J. Geol. 48, p. 785-944)
kr  P 
g 

n 
r 
kr
n
force/unit mass 
 Fluid flux mass vector (g/cm2-sec)
 k
 rock (matrix) permeability (cm2)
 r
 fluid density (g/cm3)
 [.....]  Force/unit mass acting on fluid element
 1/ n
where n Kinematic Viscosity
= mr cm2/sec
Rewrite Darcy's Law:
qv
Hubbert (1940;
J. Geol. 48, p. 785-944)
k  P 

g 

n 
r 
k

rg  P 
rn



k
rn
force/unit vol 
qv  Fluid volumetric flux vector (cm3/cm2-sec) = qm r
 k
 rock (matrix) permeability (cm2)
 [.....]  Force/unit vol. acting on fluid element
 1/ n
where n Kinematic Viscosity
= mr cm2/sec
STATIC FLUID (NO FLOW)
qm 
kr  P 
g 

n 
r 
Force/unit mass = 0
∂P/∂z = rg
for 
qm =0
∂P/∂x =0
∂P/∂y = 0
Converse:
Horizontal pressure gradients require fluid flow
STATIC FLUID (NO FLOW)
qm 
0
kr  P 
g 

n 
r 
Force/unit mass = 0
∂P/∂z = rg
for 
qm =0
∂P/∂x =0
∂P/∂y = 0
Converse:
Horizontal pressure gradients require fluid flow
Darcy's Law: Isotropic Media:
OK only if Kx = Ky = Kz
q = - K h
Darcy's Law: Anisotropic Media
K is a tensor
Simplest case (orthorhombic?)
where principal directions of anisotropy coincide with x, y, z

q =–
K xx 0 0
0 Kyy 0
0 0 K zz
Thus
i
q x = – Kxx h
x
j
qy = – Kyy h
y
h
i
x
j h
y
k h
z
qz = –
k
Kzz h
z
General case: Symmetrical tensor
Kxy =Kyx
Kzx=Kxz
Kyz =Kzy

q =–
K xx K xy Kxz
K yx K yy Kyz
K zx K zy Kzz
h
–
K
q x = xx x

i h
x
j h
y
k h
z
h
– K xy h
–
K
xz z
y
h – K h – K h
–
K
q y = yx x
yy y
yz z
h
h
h
–
K
–
K
–
K
q z = zx x
zy y
zz z
End
Relevant Physical Properties for Darcy’s Law
Hydraulic conductivity
Density
Kinematic Viscosity
Dynamic Viscosity
Porosity
Permeability
qv
qv 
K kg/n
r
n
mnr
f
k
= - Kh
k
rn
qm  r q v
rg  P
cm/s
g/cm3
cm2/sec
poise
dimensionless
cm2
g/cm3
DENSITY (r)
also, Specific weight (weight density)
gr g
r = f(T,P)
rT,P  ro 1 –  (T–To) +(P–Po)
for small ,
where
Thermal expansivity
 1

V
Isothermal Compressibility
V
T
=
P
1
–
r

1
T  –
V
r
T
because
P
V
=
P T
1
r
dr
dV
=–
r
V
r
P
T
DYNAMIC VISCOSITY
m
A measure of the rate of strain in an imperfectly elastic material
subjected to a distortional stress.
For simple shear t = m ∂u∂y
Units (poise; 1 P = 0.1 N sec/m2 = 1 dyne sec/cm2
Water 0.01 poise (1 centipoise)
KINEMATIC VISCOSITY
n
 n = mr m2/sec or cm2/sec
Water:
10-6 m2/sec
Basaltic Magma
0.1 m2/sec
= 10-2 cm2/sec
Asphalt @ 20°C
or granitic magma 102 m2/sec
Mantle
1016 m2/sec
see Tritton p. 5; Elder p. 221)
Darcy's Law:
Hubbert (1940;
k  P 
 g 
 
n 
r 
qv
-
kg
n
J. Geol. 48, p. 785-944)
h
  Kh
where:

q
v
 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
POROSITY (f,or n)
dimensionless
Ratio of void space to total volume of material
f
= Vv/VT
Dictates how much water a saturated material can contain
Important influence on bulk properties of material
e.g., bulk r, heat cap., seismic velocity……
Difference between Darcy velocity and average microscopic velocity
Decrease with depth:
Shales
Sandstones:
f = foe-cz
f = fo - c z
exponential
linear
FCC BCC
26% 32%
Gravel
Shale
Sandstone
0
8
Sand
Silt &
Clay
Siltstone
Limestone
& Dolostone
Fractured
crystalline
rocks
Simple cubic
47.6%
karstic

Pumice
Basalt
16
24
32
40
Porosity, %
48
56
64
Shales
(Athy,
1930)
Sandstones
(Blatt, 1979)
Domenico & Schwartz (1990)
PERMEABILITY (k)
units cm2
Measure of the ability of a material to transmit fluid
under a hydrostatic gradient
Most important rock parameter pertinent to fluid flow
Relates to the presence of fractures and interconnected voids
1 darcy  0.987 x 10-8 cm2
0.987 x 10-12 m2
(e.g., sandstone)
Approximate relation between K and k
Km/s  107 k m2  10-5 kdarcy
1nd
1md
1 md
Clay
1d
Silt
Sand
Gravel
Sandstone
B
Shale
1000d
Limestone
argillaceous
cavernous
Basalt
Crystalline Rocks
-18
10
-16
10
-14
10
-12
10
-10
10
-8
10
-6
10
PERMEABILITY, cm
2
-4
10
-2
10
GEOLOGIC REALITIES OF PERMEABILITY (k)
Huge Range in common geologic materials > 1013 x
Decreases super-exponentially with depth
k = Cd2
k = a3/12L
for granular material,
where d = grain diameter, C is complicated parameter
for parallel fractures of aperture width “a” and spacing L
k is dynamic (dissolution/precipitation, cementation,
thermal or mechanical fracturing; plastic deformation)
Scale dependence: kregional ≥ kmost permeable parts of DH >> klab; small scale
)
MEANS: (D&S, p. 66-70)
Arithmetic Mean
M = S Xi/N
Geometric Mean
G = {X1 X2 X3 .....XN}1/N
Harmonic Mean
H = N/ S (1Xi)
Commonly (always?) ,
M>G>H
Example:
N = 3 samples:
Xi = 2, 4, 8
M = 4.6667
G = 4.0
H = 3/(7/8) = 3.428
Xi = data points, N = # samples
PERMEABILITY ANISOTROPY
Both Hyd raulic Conduc tivit y (K) and the Permeabili ty (k) can be anis otropic.
Important case: Laye red stratigraphic sequence, e .g., interbedded sst & shales
Horizon tal bulk hyd. conduc tivit y:
Vertical bulk hyd rauli c condu ctivit y:
Kx = SmiKi / Smi where m is layer thickness
Kz = Smi/ S(mi/Ki)
=> For horizont al flow, the most permeable un it s dominate, but
For vertical flow, the least permeable un it s domi nate!
Anisotropy Ratio: Kx / K z ~ 1 to 10x, for typical l aye r
(e.g., becau se of pr eferred orientation, schistosit y...)
Anisotropy Ratio: Kx / K z to 10 6 or more, for stratigraphic sequen ce
In general, both K and k are tensors, and the direction of fluid
flow need not coincide with the gradient in hydraulic head
Stratigraphic Sequence
Kx > Kz
Horizontal Flow
Q   Qi   K total (1  mi )h
where (1  mi )  A
 mi  total sequence thickness
1
= unit width out of page
Q = K1 (1 m1 )h  K 2 (1 m2 )h  K 3 (1 m 3 )h + .....
=  ( K i mi )h
So:
 (mi Ki )
Kx 
 mi
Horizontal K is simple mean, weighted by layer thickness
Stratigraphic Sequence
 (mi Ki )
Kx 
 mi
Vertical Flow thru Stratigraphic Sequence
q  q1  q2  q 3  ... Same flow thru each layer
q = - K totalh  - K totalh /l  - K totalh /  mi
qi   K i hi /mi
and
So
so
hi   qmi /K i
htotal   hi
Kz 
 mi
 (mi / Ki )
Kz is Harmonic Mean, weighted by layer thickness
Stratigraphic Sequence
Kz 
Kx 

 (mi Ki )
 mi
 mi
 (mi / Ki )
PERMEABILITY ANISOTROPY
Justification: For vertical flow, Flux must be the same thru each layer!
(see F&C, p. 33-34)
q = Kz,bulk (∆h/m)
= K1 (∆h1/m1) = K2 (∆h2/m2) = ....... = Kn (∆hn/mn)
=> Kz,bulk = q m/ ∆h = q m/ (∆h1 + ∆h2 + .... + ∆hn)
= q m/ (q m1/K1 + q m2/K2 + .... + q mn/Kn) =
= m / S(mi/Ki )
=> For horizontal flow, the most permeable units dominate, but
For vertical flow, the least permeable units dominate!
Anisotropy Ratio: Kx / Kz ~ 1 to 10x, for typical layer
(e.g., because of preferred orientation, schistosity...)
Anisotropy Ratio: Kx / Kz up to 106 or more, for stratigraphic sequence
In general, for layered anisotropy: Kx > Kz
However, for fracture-related anisotropy, commonly Kz > Kx
End
Aquifers
Saturated geologic formations with sufficient porosity fand permeability k
to allow significant water transmission under ordinary hydraulic gradients.
Normally, k ≥ 0.01 d
e.g., Unconsolidated sands & gravels; Sandstone, Limestone,
fractured volcanics & fractured crystalline rocks
Aquitard
Geologic formations with low permeability that can store ground water
and allow some transmission, but in an amount insufficient for production.
Less permeable layers in stratigraphic sequence;
= Leaky confining layer
e.g., clays, shales, unfractured crystalline rocks
Aquiclude
Saturated geologic unit incapable of transmitting significant water
Rare.
Unconfined Aquifer: aquifer in which the water table forms upper boundary.
= water table aquifer
e.g., Missouri R.; Mississippi R., Meramec River valleys
Hi yields, good quality
e.g., 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
after Driscoll, FG (1986)
http://www.uwsp.edu/water/portage/undrstnd/aquifer.htm
Unconfined Aquifer
after Fetter
http://www.uwsp.edu/water/portage/undrstnd/aquifer.htm
Perched and
Unconfined Aquifers
after Fetter
http://www.uwsp.edu/water/portage/undrstnd/aquifer.htm
Confined Aquifer
after Fetter
http://www.uwsp.edu/water/portage/undrstnd/aquifer.htm
Hubbert (1940)
Black
Hills
Potentiomtric Surface, Dakota Aquifer
after Darton 1909
Unconfined Aquifer: Water table aquifer
Aquifer in which the water table forms upper boundary.
e.g., MO, Miss, Meramec River valleys. Hi yields, good quality
e.g., Ogalalla Aquifer (High Plains aquifer)
Properties:
1) Get large production for a given head drop, as Specific Yield Sy is large (~0.2
2) Storativity S = Sy + Ss*h  Sy, commonly
(eq 4.33 Fetter)
3) Easily contaminated
4) Artesian flow possible
Confined Aquifer: Aquifer between two aquitards.
Artesian aquifer if the water level in a well rises above aquifer
Flowing Artesian aquifer if the water level in the well
rises above the ground surface.
e.g., Dakota Sandstone
Properties:
1) Get large changes in pressure (head) with ~ no change in the thickness
of the saturated column. Potentiometric sfc remains above the unit.
2) Get large head drop for a given amount of production, as Ss is very small.
3) Storativity S= Ss*m where Ss = specific storage
Commonly, S ~ 0.005 to 0.0005 for aquifers
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 Fg
GM
Fg 
r
Gravitational Potential Fg
GM
Fg 
r
GM
F g   2  Force
r
 F g  4  Gr
2
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
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)
Flow beneath Dam
Vertical x-section
Flow toward Pumping Well,
next to river
Plan view
River Channel
Domenico &
Schwartz(1990)
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)
Hubbert 1957
76.1 mi2
5000
Broad Run, Leesburg, VA
4000
Qcalc = 4580*Q(0.2)
Q, cfs
3000
2000
1000
0
12
14
16
YearDay 2005
18
20
22
15.3
Broad Run, VA
Recession 80 to 25%
YearDay 2005
15.2
t = b*4580/Q
+t
15.1
ref
15
14.9
14.8
y = 14.566 + 0.16633x R= 0.99689
14.7
1
1.5
2
2.5
4580/Q
3
3.5
4
9
Broa d Run
8
ln(Q)
7
6
5
4
y = 14.015 - 0.49664x R= 0.88317
3
12
14
16
18
YearDay 2005
20
22
5000
Broad Run
4000
Q=1343-796.44 S +123.31 S
2
R=.9996
Q, cfs
3000
2000
1000
0
2
3
4
5
6
Stage, ft
7
8
9
10
7000
Jacks Fork
6000
Q = 786.8 -582.6 S +137.62 S
R=.99979
5000
2
Q (cfs)
4000
3000
2000
1000
0
2
3
4
5
6
Stage (ft)
7
8
9
14.8
Jacks Fork
y = 13.798 + 0.22077x R= 0.99981
DATE Jan 05
14.6
14.4
14.2
14
13.8
13.6
0
0.5
1
1.5
2
Qp/Q
2.5
3
3.5
4
7000
Jacks Fork
6000
Q (USGS)
Q(0.35)
Q (cfs)
5000
4000
3000
2000
1000
0
5
10
15
20
DATE Jan 05
25
30
35
7000
0
0.2
300+6040Q(.35)
Q (cfs)
5000
0.4
4000
3000
2000
1000
0
8
10
12
14
16
DATE Jan 05
18
20
22
24
ppt (in)
JacksFork
Q (data) in MO
6000
7000
Jacks Fork
6000
Q (USGS)
Q(0.35)
Q (cfs)
5000
4000
3000
2000
1000
0
5
10
15
20
DATE Jan 05
25
30
35
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
Fetter, after Toth (1963)
Ghyben-Herzberg
r fw
z  hf
 40h f
r sw  r fw

hf
z
Salt Water
P
Sea level
Fresh Water