FLUID STATICS No flow Surfaces of const P and coincide along gravitational equipotential surfaces h...

FLUID STATICS No flowSurfaces of const P and coincide along gravitational equipotential surfaces

€

dP

dz= −ρg

P = ρg(h − z) for constant ρ

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

-30

-25

-20

-15

-10

-5

0 0

0.2

0.4

0.6

0.8

1

DEPTH, m

Distance

P, bar

P = 0.1 bar/m

0.6

1.2

2.4

1.8

3.0

-30

-25

-20

-15

-10

-5

0 0

0.2

0.4

0.6

0.8

1

DEPTH, m

Distance

P, bar

-30

-25

-20

-15

-10

-5

0 0

0.2

0.4

0.6

0.8

1

DEPTH, m

Distance

P, barh

PL > Ph Ph = 0.1 bar/m

0.6

1.2

1.8

2.4

3.0

FLUID DYNAMICS in PERMEABLE MEDIA

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 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 TexasFlowing 100 yearsHot, sulfur-rich, artesian water

http://www.texasescapes.com/TexasGulfCoastTowns/Humble-Texas.htm

-30

-25

-20

-15

-10

-5

0 0

0.2

0.4

0.6

0.8

1

DEPTH, m

Distance

P, bar

P = 0.1 bar/m

0.6

1.2

1.8

2.4

3.0

-30

-25

-20

-15

-10

-5

0 0

0.2

0.4

0.6

0.8

1

DEPTH, m

Distance

P, bar

P = 0.1bar/m

0.6

1.2

1.8

2.4

3.0

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 SystemsFluid 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 "Darcy Velocity" where 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

q = - 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

Negative sign: flow is down gradient

Actual microscopic velocity ()

= q/ = Darcy Velocity/effective porosity

Clearly, > q

HYDRAULIC CONDUCTIVITY, K m/s

K = kg/ kg/ units of velocity

Proportionality constant in Darcy's Law Property of both fluid and medium

see D&S, p. 62

HYDRAULIC POTENTIAL (): energy/unit mass cf. h = energy/unit weight

= g h = gz + P/w

Consider incompressible fluid element @ elevation zi= 0 pressure Pi i and velocity v = 0

Move to new position z, P, v

Energy difference: lift mass + accelerate + compress (= VdP) = mg(z- zi) + mv2/2 + m V/m) dP

latter term = m (1/dP

Energy/unit mass g z + v2/2 + /dP

For incompressible fluid = const) & slow flow (v2/2 0), zi=0, Pi = 0

Energy/unit mass: g z + P/ = g h

Force/unit mass = = g - P/

Force/unit weight = h = 1 - P/g

Rewrite Darcy's Law: Hubbert (1940, J. Geol. 48, p. 785-944)

qm Fluid flux mass vector (g/cm2-sec) k rock (matrix) permeability (cm2) fluid density (g/cm3) [.....] Force/unit mass acting on fluid

element 1/

where Kinematic Viscosity = cm2/sec

€

qm = ρ qv = kρ

νg −

∇P

ρ

⎡

⎣ ⎢

⎤

⎦ ⎥

= kρ

νforce/unit mass[ ]

€

qv = k

νg −

∇P

ρ

⎡

⎣ ⎢

⎤

⎦ ⎥

= k

ρνρg − ∇P[ ]

= k

ρνforce/unit vol[ ]

Rewrite Darcy's Law: Hubbert (1940; J. Geol. 48, p. 785-944)

qv Fluid volumetric flux vector (cm3/cm2-sec) = qm

k rock (matrix) permeability (cm2) [.....] Force/unit vol. acting on fluid element 1/

where Kinematic Viscosity = cm2/sec

STATIC FLUID (NO FLOW)

€

qm = kρ

νg −

∇P

ρ

⎡

⎣ ⎢

⎤

⎦ ⎥

Force/unit mass = 0 for qm =0

∂P/∂z = g ∂P/∂x =0 ∂P/∂y = 0 Converse: Horizontal pressure gradients require fluid flow

STATIC FLUID (NO FLOW)

€

qm = kρ

νg −

∇P

ρ

⎡

⎣ ⎢

⎤

⎦ ⎥

Force/unit mass = 0 for qm =0

∂P/∂z = g ∂P/∂x =0 ∂P/∂y = 0 Converse: Horizontal pressure gradients require fluid flow

0

Darcy's Law: Isotropic Media: q = - K h OK only if Kx = Ky = Kz

Darcy's Law: Anisotropic MediaK is a tensorSimplest case (orthorhombic?) where principal directions of anisotropy coincide with x, y, z

q = –

Kxx

0 0

0 Kyy

0

0 0 Kzz

i

∂ h

∂ x

j

∂ h

∂ y

k

∂ h

∂ z

q

x

= – Kxx

∂ h

∂ x

i qy

= – Kyy

∂ h

∂ y

j qz

= – Kzz

∂ h

∂ z

k

Thus

q = –

Kxx

Kxy

Kxz

Kyx

Kyy

Kyz

Kzx

Kzy

Kzz

i

∂ h

∂ x

j

∂ h

∂ y

k

∂ h

∂ z

General case: Symmetrical tensor Kxy =Kyx Kzx=Kxz Kyz =Kzy

q

x

= – Kxx

∂ h

∂ x

– Kxy

∂ h

∂ y

– Kxz

∂ h

∂ z

qy

= – Kyx

∂ h

∂ x

– Kyy

∂ h

∂ y

– Kyz

∂ h

∂ z

qz

= – Kzx

∂ h

∂ x

– Kzy

∂ h

∂ y

– Kzz

∂ h

∂ z

Relevant Physical Properties for Darcy’s Law

Hydraulic conductivity K kg/ cm/sDensity g/cm3

Kinematic Viscosity cm2/secDynamic Viscosity poise

Porosity dimensionlessPermeability k cm2

€

qv = k

ρν ρg − ∇P[ ]

€

qm = ρ qv

qv = - Kh

DENSITY () g/cm3

also, Specific weight (weight density) g

= f(T,P)

α ≡

V

∂ V

∂ TP

= –

∂

∂ TP

because

d

= –

dV

V

Thermal expansivity

βT

≡ –

1

V

∂ V

∂ PT

=

1

ρ

∂ ρ

∂ PT

Isothermal Compressibility

ρ

T , P

≅ ρo

1 – α ( T – To

) + β ( P – Po

)

for small α , β

where

DYNAMIC VISCOSITY A measure of the rate of strain in an imperfectly elastic material

subjected to a distortional stress. For simple shear = ∂u∂y

Units (poise; 1 P = 0.1 N sec/m2 = 1 dyne sec/cm2

Water 0.01 poise (1 centipoise)

KINEMATIC VISCOSITY = m2/sec or cm2/sec

Water: 10-6 m2/sec = 10-2 cm2/sec

Basaltic Magma 0.1 m2/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; 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

POROSITY ( or n) dimensionless

Ratio of void space to total volume of material

= Vv/VT

Dictates how much water a saturated material can contain

Important influence on bulk properties of material e.g., bulk , heat cap., seismic velocity……

Difference between Darcy velocity and average microscopic velocity

Decrease with depth: Shales = oe-cz exponential

Sandstones: = o - c z linear

0 8 16 24 32 40 48 56 64

Porosity, %

Fractured Basaltcrystallinerocks

Limestone karstic & Dolostone

Shale Sandstone Siltstone

Gravel Sand Silt & Clay

FCC BCC Simple cubic 26% 32% 47.6%

Pumice

Domenico & Schwartz (1990)

Shales (Athy, 1930)

Sandstones (Blatt, 1979)

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 .987 x 10-12 m2 (e.g., sandstone)

Approximate relation between K and k Km/s 107 k m

2 10-5 kdarcy

2

10 10 10 10 10 10 10 10 10-18 -16 -14 -12 -10 -8 -6 -4 -2

PERMEABILITY, cm

1nd 1d 1 md 1 d 1000d

Clay Silt Sand Gravel

Shale Sandstone

argillaceous Limestone cavernous

Basalt

Crystalline Rocks

GEOLOGIC REALITIES OF PERMEABILITY (k)

Huge Range in common geologic materials > 1013 x

Decreases super-exponentially with depth

k = Cd2 for granular material, where d = grain diameter, C is complicated parameter

k = a3/12L 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 = Xi/N Xi = data points, N = # samples

Geometric Mean G = {X1 X2 X3 .....XN}1/N

Harmonic Mean H = N/ 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

In general, both K and k are tensors, and the direction of fluid flow need not coincide with the gradient in hydraulic head

PERMEABILITY ANISOTROPY

Both Hydraulic Conductivity (K) and the Permeability (k) can be anisotropic.

Important case: Layered stratigraphic sequence, e.g., interbedded sst & shales

Horizontal bulk hyd. conductivity: Kx = miK i / mi w here mis l ayerthickness

Vertica l bulk hydrauli c conductivity: Kz = mi/ (mi/K i)

=> For horizontal fl ,ow t hemost permeabl e units dominate, but For vertical flow, the least permeabl e units dominate!

Anisotropy Ratio: Kx / Kz ~ t o x, for typica l layer( .e g., becaus e of preferr ed orientation, schistosity...)

Anisotropy Ratio: Kx / Kz to 6 ormore, for stratigraphic sequence

Stratigraphic Sequence

Kx > Kz

€

Kx =miKi( )∑mi∑

€

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 − K2 1×m2( )∇h − K3 1×m3( )∇h + .....

= − Kimi∑( )∇h

So:

Horizontal K is simple mean, weighted by layer thickness

Horizontal Flow

€

Kx =miKi( )∑mi∑


€

Kz =mi∑

mi / Ki( )∑

€

q = q1 = q2 = q3 = ... Same flow thru each layer

q = - K total∇h = - K totalΔh /Δl = - K totalΔh / mi∑

€

qi = − KiΔhi /mi so Δhi = −qmi/Ki

and Δhtotal = Δhi∑

So

Vertical Flow thru Stratigraphic Sequence

Kz is Harmonic Mean, weighted by layer thickness

€

Kx =miKi( )∑mi∑

€

Kz =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 / 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

AquifersSaturated geologic formations with sufficient porosity 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

AquitardGeologic 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

AquicludeSaturated 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

after Fetterhttp://www.uwsp.edu/water/portage/undrstnd/aquifer.htm

Unconfined Aquifer


Perched and Unconfined Aquifers


Confined Aquifer

Hubbert (1940)

after Darton 1909

Potentiomtric Surface, Dakota Aquifer

BlackHills

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 (~0.25).

2) Storativity S = Sy + Ss*h Sy, commonly (eq 4.33 Fetter)

3) Easily contaminated4) Artesian flow possible

Confined Aquifer: Aquifer between two aquitards. Artesian aquifer if the water level in a well rises above aquiferFlowing 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; 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

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

Fetter

No Flow

No

Flow

No Flow

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 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 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)

Domenico & Schwartz(1990)

Flow beneath DamVertical x-section

Flow toward Pumping Well,next to riverPlan view

River Channel

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

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)

Hubbert 1957

76.1 mi2

0

1000

2000

3000

4000

5000

12 14 16 18 20 22

Broad Run, Leesburg, VA

Q, cfs

YearDay 2005

Qcalc = 4580*Q(0.2)

14.7

14.8

14.9

15

15.1

15.2

15.3

1 1.5 2 2.5 3 3.5 4

Broad Run, VARecession 80 to 25%

y = 14.566 + 0.16633x R= 0.99689

4580/Q

t = b*4580/Q + tref

3

4

5

6

7

8

9

12 14 16 18 20 22

Broad Run

y = 14.015 - 0.49664x R= 0.88317

ln(Q)

YearDay 2005

0

1000

2000

3000

4000

5000

2 3 4 5 6 7 8 9 10

Broad Run

Q, cfs

Stage, ft

Q=1343-796.44 S +123.31 S2 R=.9996

0

1000

2000

3000

4000

5000

6000

7000

2 3 4 5 6 7 8 9

Q = 786.8 -582.6 S +137.62 S2 R=.99979

Q (cfs)

Stage (ft)

Jacks Fork

13.6

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Jacks Fork y = 13.798 + 0.22077x R= 0.99981

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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

Fetter, after Toth (1963)

Ghyben-Herzberg

Salt Water

Fresh Water

hf

z

€

z = h fρ fw

ρ sw −ρ fw

≈ 40h f

Ghyben-Herzberg

P

Sea level

FLUID STATICS No flow Surfaces of const P and coincide along gravitational equipotential surfaces h...

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