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