Darcy Lab: Describe Apparatus

Q = K A ∂h/∂x

cm3/sec = cm/sec cm2 cm/cm

after Domenico & Schwartz (1990)

Flow toward Pumping Well,next to river = line source

= constant head boundary

Plan view

River C

hannel

Line S

ource

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

Flow Net Rules:

Flowlines are perpendicular to equipotential lines (isotropic case)

Spacing between equipotential lines L:If spacing between lines is constant, then K is constantIn general K1 m1/L1 = K2 m2/L2 where m = x-sect thickness of aquifer;

L = distance between equipotential lines

For layer of const thickness, K1/L1 ~ K2/L2

No Flow BoundariesEquipotential lines meet No Flow boundaries at right anglesFlowlines are tangent to such boundaries (// flow)

Constant Head BoundariesEquipotential lines are parallel to constant head boundaries Flow is perpendicular to constant head boundary

Impermeble

Boundary

Constant Head

Boundary

Water Table

Boundary

after Freeze & Cherry

FLOW NETS

http://photos.aip.org/

MK Hubbert1903-1989

MK Hubbert (1940)http://www.wda-consultants.com/java_frame.htm?page17

MK Hubbert (1940)http://www.wda-consultants.com/java_frame.htm?page17

Consider piezometers emplaced near hilltop & near valley

Fetter, after Hubbert (1940)

Fetter, after Hubbert (1940)

Cedar Bog, OH

Piezometer Cedar Bog, Ohio

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

Bluegrass Spring

Criss

MK Hubbert (1940)http://www.wda-consultants.com/java_frame.htm?page17

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)

How do we know basic flownet picture is correct?

How do we know basic flownet picture is correct?Mathematical solutions (Toth, 1962, 1963)Numerical Simulations Data

Basin Geometry: Sinusoidal water table on a regional topo slope Toth (1962, 1963)

h(x, z0) = z0 + Bx/L + b sin (2x/)

constant + regional slope + local relief

Sinusoidal Water Table with a Regional Slope

Z

Distance, x

Z = Z0

X = X0

X = L

B

Basin Geometry: Sinusoidal water table on a regional topo slope Toth (1962, 1963)

h(x, z0) = z0 + Bx/L + b sin (2x/)

constant + regional slope + local relief

Solve Laplace’s equation

Simulate nested set of flow systems

€

∇2h = 0

e.g., D&S

€

Φ x , z( ) = A'− B' Cosh π z /L[ ] Cos π x /L[ ]

Cosh π z0 /L[ ]

How do we get q?

Regional flow pattern in an area of sloping topography and water table.Fetter, after Toth (1962) JGR 67, 4375-87.

No Flow

No

Flow

No Flow

Discharge Recharge

after Toth 1963Australian Government

Local

Flow

Systems

Intermediate Flow System

RegionalFlow System

Conclusions

General slope causes regional GW flow system, If too small, get only local systems

If the regional slope and relief are both significant, get regional, intermediate, and local GW flow systems.

Local relief causes local systems. The greater the amplitude of the relief, the greater the proportion of the water in the local system

If the regional slope and relief are both negligible, get flat water table often with waterlogged areas mostly discharged by ET

For a given water table, the deeper the basin, the more important the regional flow

High relief & deep basins promote deep circulation into hi T zones

End 24Begin 25

Hubbert (1940)

MK Hubbert1903-1989

http://www.wda-consultants.com/java_frame.htm?page17

Equipotential Line

Flow Line

FLOW NETS

AIP

How do we know basic flownet picture is correct?DataMathematical solutions (Toth, 1962, 1963)Numerical Simulations

Piezometer Cedar Bog, Ohio

Regional flow pattern in an area of sloping topography and water table.Fetter, after Toth (1962) JGR 67, 4375-87.

No Flow

No

Flow

No Flow

Discharge Recharge

Pierre Simon Laplace

1749-1827

€

Φ x , z( ) = A'− B' Cosh π z /L[ ] Cos π x /L[ ]

Cosh π z0 /L[ ]

€

∇2h = 0

Numerical Simulations

Basically reproduce Toth’s patterns

High K layers act as “pirating agents

Refraction of flow lines tends to align flow parallel to hi K layer, and perpendicular to low K layers

after Freeze and Witherspoon 1967http://wlapwww.gov.bc.ca/wat/gws/gwbc/!!gwbc.html

Effect of Topography on Regional Groundwater Flow

Isotropic Systems

Regular slope

Sinusoidal slope

Isotropic Aquifer

Anisotropic Aquifer Kx: Kz = 10:1

after Freeze *& Witherspoon 1967

after Freeze *& Witherspoon 1967

Layered Aquifers

after Freeze *& Witherspoon 1967

Confined Aquifers

Sloping Confining Layer

Horizontal Confining Layer

Conclusions

General slope causes regional GW flow system, If too small, get only local systems

Local relief causes local systems. The greater the amplitude of the relief, the greater the proportion of the water in the local system

If the regional slope and relief are both negligible, get flat water table often with waterlogged areas mostly discharged by ET

If the regional slope and relief are both significant, get regional, intermediate, and local GW flow systems.

For a given water table, the deeper the basin, the more important the regional flow

High relief & deep basins promote deep circulation into hi T zones

Flow in a Horizontal Layers

Case 1: Steady Flow in a Horizontal Confined Aquifer

€

q = QA

= − K ∇h

€

Q' = − K m ∇h Flow/ unit width:

Darcy Velocity q:

Typically have equally-spaced equipotential lines

Case 2: Steady Flow in a Horizontal, Unconfined Aquifer

€

Q' = − K h ∇h = −K 2

∇h2 Flow/ unit width:m2/s

Dupuit (1863) Assumptions:Grad h = slope of the water tableEquipotential lines (planes) are verticalStreamlines are horizontal

Q’dx = -K h dh

€

Q'dx0

L∫ = − K h dh

h1

h2

∫

€

Q'L = −K2

h22 - h1

2( )

€

Q' = −K2

h2

2 - h12

L

⎛

⎝ ⎜

⎞

⎠ ⎟ Dupuit

Equation Fetter p. 164

0

5

10

15

20

0 5 10 15 20

Dupuit eq.

Distance x, m

K = 10-5 m/s

Q' = 8 x10-5 m2/s

Impervious Base

€

∇

h

€

h = h12 −

2Q' x K

⎛ ⎝ ⎜

⎞ ⎠ ⎟0.5

cf. Fetter p. 164

Steady flowNo sources or sinks

cf. Fetter p. 167 F&C 189

Q’ = -K h dh/dx

dQ’/dx = 0 continuity equation

€

d2h2

dx2 = 0

€

d2h2

dx2 +d2h2

dy2 = 0 = ∇2h2

So:

More generally, for an Unconfined Aquifer:

Steady flowwith source term:Poisson Eq in h2

where w = recharge cm/sec

Steady flow:No sources or sinksLaplace’s equation in h2

Better Approach

€

-K 2

∇2h2 = w

for one dimensional flow

€

-K 2

∇2h2 = w Steady unconfined flow:with a source termPoisson Eq in h2

€

-K 2

∂2h2

∂x2 = w 1-D

Solution:

€

h2 = -wx2

K+ Ax + B

Boundary conditions: @ x= 0 h= h1 ; @ x= L h= h2

€

h2 = w L − x( )x

K-

h12 − h2

2( )x

L+ h1

2 cf. Fetter p. 167 F&C 189

cf. Fetter p. 167 F&C 189

w

Unconfined flow with recharge

5

10

15

20

25

30

-200 0 200 400 600 800 1000 1200

Distance x, m

w = 10-8 m/sK = 10-5 m/s@ x=0 h1 = 20m

@ x=1000m h2 = 10m

€

h2 = w L − x( )x

K-

h12 − h2

2( )x

L+ h1

2

Finally, for unsteady unconfined flow: Boussinesq Eq.

€

∂∂x

h∂h∂x

⎛ ⎝ ⎜

⎞ ⎠ ⎟ +

∂∂y

h∂h∂y

⎛

⎝ ⎜

⎞

⎠ ⎟=

Sy

T∂h∂t

Sy is specific yield

Fetter p. 150-1

For small drawdown compared to saturated thickness b:Linearized Boussinesq Eq. (Bear p. 408-9)

€

∂2h∂x2 +

∂2h∂y2 =

Sy

Kb∂h∂t

€

∇2h = AT

€

∇2h = 0

€

∇2h = ST

∂h∂t

Laplace’s EquationSteady flow

Poisson’s EquationSteady Flow with Source or Sink

Diffusion Equation

End Part II

Pierre Simon Laplace1749-1827

Dibner Lib.

http://upload.wikimedia.org/wikipedia/en/f/f7/Hubbert.jpg

MK Hubbert1903-1989

wikimedia.org

Leonhard Euler1707 - 1783

http://photos.aip.org/

Charles V. Theis19-19

€

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

After Toth 1983

after Johnson 1975

€

∂ 2h

∂ r 2 + 1

r

∂h

∂r =

1

D

∂h

∂t

Radial flow

Initial Condition & Boundary conditions:

€

h(r , 0 )= h0 h(∞, t )= h0 limr→0

r∂h∂r

⎛ ⎝ ⎜

⎞ ⎠ ⎟=

Q2πT

for t > 0

Transient flow, Confined Aquifer, No rechargeConstant pumping rate Q

€

∂ 2h

∂ r 2 + 1

r

∂h

∂r =

1

D

∂h

∂t

Radial flow

Initial Condition & Boundary conditions:

€

h(r , 0 )= h0 h(∞, t )= h0 limr→0

r∂h∂r

⎛ ⎝ ⎜

⎞ ⎠ ⎟=

Q2πT

for t > 0

€

W (u )= −Ei(−u )=e−ξ

ξu

∞∫ dξ where u =

r2S4tT

=r2

4Dt

and where

Solution: “Theis equation” or “Non-equilibrium Eq.”

€

W ∞( ) = 0 W 0( ) = ∞

€

Drawdown = h0 − h =Q

4πTW (u )

where

Approximation for t >> 0

€

Drawdown = h0 − h ≅Q

4πTln

2.25 D t r2

⎛ ⎝ ⎜

⎞ ⎠ ⎟

D&S p. 151

€

W (u )= −Ei(−u )= −0.577216 − lnu + u −u2

4+

u3

3× 3!−

u4

4 × 4!+

u5

5 × 5!− ....

€

W (u ) ≅ −0.577216 − lnu for small u < 0.1 ; i.e., long times or small r

USGS Circ 1186

Pumping of Confined Aquifer

Not GW “level” Potentiometric sfc!

USGS Circ 1186

Pumping of Unconfined Aquifer

USGS Circ 1186

Santa Cruz RiverMartinez Hill,South of Tucson AZ

1989>100’ GW drop

1942Cottonwoods,Mesquite

€

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

USGS Circ 1186

Pumping @ rate Q1

(note divide)

Initial Condition

Pumping @ rate Q2 >Q1

-2

0

2

4

6

8

0 0.5 1 1.5 2

Well Function W(u)= - Ei (-u)

W(u)

u

W(u) ~ -0.577216 - ln(u) OK for u < 0.1

Domenico & Schwartz (1990)

Flow beneath DamVertical x-section

Flow toward Pumping Well,next to river = line source

= constant head boundary

Plan view

River Channel

after Toth 1963http://www.co.portage.wi.us/Groundwater/undrstnd/topo.htm

after Toth 1963Australian Government

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

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

Clarence King

1st Director of USGS

1879-1881