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Fetter, Ch. 4. 4.1 Introduction 4.2 Mechanical Energy 4.3 Hydraulic Head 4.4 Head in Water of Variable Density 4.5 Force Potential and Hydraulic Head 4.6 Darcy’s Law 4.7 Equations of Groundwater Flow 4.8 Solutions of Flow Equations 4.9 Gradient of Hydraulic Head

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fetter ch 4
Fetter, Ch. 4
  • 4.1 Introduction
  • 4.2 Mechanical Energy
  • 4.3 Hydraulic Head
  • 4.4 Head in Water of Variable Density
  • 4.5 Force Potential and Hydraulic Head
  • 4.6 Darcy’s Law
  • 4.7 Equations of Groundwater Flow
  • 4.8 Solutions of Flow Equations
  • 4.9 Gradient of Hydraulic Head
  • 4.10 Relationship of Flow to Gradient
  • 4.11 Flow Lines and Flow Nets
  • 4.12 Refraction of Flow Lines
  • 4.13 Steady Flow in Confined Aquifers
  • 4.14 Steady Flow in Unconfined Aquifers
introduction
Introduction
  • Groundwater possesses energy
  • in a variety of forms:
  • Mechanical
  • Thermal
  • Chemical
fetter ch 41
Fetter, Ch. 4
  • 4.1 Introduction
  • 4.2Mechanical Energy
  • 4.3 Hydraulic Head
  • 4.4 Head in Water of Variable Density
  • 4.5 Force Potential and Hydraulic Head
  • 4.6 When is Darcy’s Law Applicable?
  • 4.7 Equations of Groundwater Flow
  • 4.8 Solutions of Flow Equations
  • 4.9 Gradient of Hydraulic Head
  • 4.10 Relationship of Flow to Gradient
  • 4.11 Flow Lines and Flow Nets
  • 4.12 Refraction of Flow Lines
  • 4.13 Steady Flow in Confined Aquifers
  • 4.14 Steady Flow in Unconfined Aquifers
head in water of variable density
Head in Water of Variable Density

If z1 = z2 ,

hf = (ρp/ρf ) hpoint

head in water of variable density1
Head in Water of Variable Density

Which way is water moving vertically—

up or down?

(see p.120)

slide6

Head in Water of Variable Density

hfresh = 5,500 ft

ρ = 1,000 kg/m3

5,000 ft

ρ = 1,100 kg/m3

hpoint =5,000 ft

Z =0 ft

fetter ch 42
Fetter, Ch. 4
  • 4.1 Introduction
  • 4.2 Mechanical Energy
  • 4.3 Hydraulic Head
  • 4.4 Head in Water of Variable Density
  • 4.5 Force Potential and Hydraulic Head
  • 4.6 When is Darcy’s Law Applicable?
  • 4.7 Equations of Groundwater Flow
  • 4.8 Solutions of Flow Equations
  • 4.9 Gradient of Hydraulic Head
  • 4.10 Relationship of Flow to Gradient
  • 4.11 Flow Lines and Flow Nets
  • 4.12 Refraction of Flow Lines
  • 4.13 Steady Flow in Confined Aquifers
  • 4.14 Steady Flow in Unconfined Aquifers
fetter ch 43
Fetter, Ch. 4
  • 4.1Introduction
  • 4.2Mechanical Energy
  • 4.3 Hydraulic Head
  • 4.4 Head in Water of Variable Density
  • 4.5 Force Potential and Hydraulic Head
  • 4.6 When is Darcy’s Law Applicable?
  • 4.7 Equations of Groundwater Flow
  • 4.8 Solutions of Flow Equations
  • 4.9 Gradient of Hydraulic Head
  • 4.10Relationship of Flow to Gradient
  • 4.11 Flow Lines and Flow Nets
  • 4.12 Refraction of Flow Lines
  • 4.13 Steady Flow in Confined Aquifers
  • 4.14 Steady Flow in Unconfined Aquifers
slide11

Relationship of flow to gradient

Hydraulic conductivity ellipse: technique for determining effect of anisotropy on flow

Look at result: is flow deflected towards or away from highest K?

slide12

Relationship of flow to gradient

Flow nets in isotropic (A) and anisotropic (B) mediums

slide13

Relationship of flow to gradient

Flow nets in isotropic (A) and anisotropic (B) mediums

slide14

Flow Nets

El. = 200 ft

A

F

B

G

H

E

D

C

30 ft

El. = 200 ft

30 ft

A

F

B

G

H

E

D

C

fetter ch 44
Fetter, Ch. 4
  • 4.1Introduction
  • 4.2Mechanical Energy
  • 4.3 Hydraulic Head
  • 4.4 Head in Water of Variable Density
  • 4.5 Force Potential and Hydraulic Head
  • 4.6 When is Darcy’s Law Applicable?
  • 4.7 Equations of Groundwater Flow
  • 4.8 Solutions of Flow Equations
  • 4.9 Gradient of Hydraulic Head
  • 4.10 Relationship of Flow to Gradient
  • 4.11 Flow Lines and Flow Nets
  • 4.12 Refraction of Flow Lines
  • 4.13 Steady Flow in Confined Aquifers
  • 4.14 Steady Flow in Unconfined Aquifers
slide16

El. = 200 ft

A

F

B

G

H

E

C

30 ft

El. = 200 ft

30 ft

A

Each stream tube carries

equal flow

F

B

El. = 170 ft

G

H

E

195

D

C

190

185

slide17

El. = 200 ft

A

F

B

G

H

E

D

C

30 ft

El. = 200 ft

30 ft

A

F

B

El. = 170 ft

G

H

E

195

D

C

At point C:

ht = 192.5 ft

z = 128 ft

hp = 64.5 ft

p = 64.5 ft x 62.4 pcf

= 4,025 psf

190

185

slide18

Flow Nets

El. = 200 ft

30 ft

El. = 200 ft

30 ft

A

Each stream tube carries

equal flow

F

B

El. = 170 ft

G

H

E

full

stream

tube

ns= 2.2

nd = 6

ns / nd = 2.2 / 6 = 0.37

Q = K grad h A

Q for square #1 = Q1

Q1 = K (grad h) A

full

stream

tube

195

D

C

partial

stream

tube

190

185

slide19

El. = 200 ft

A

  • Flow net properties:
  • Flow and equipotential lines form squares,
  • Each stream tube carries equal flow.
  • Partial flow tubes are allowed.

30 ft

El. = 200 ft

30 ft

A

F

B

El. = 170 ft

G

H

E

195

D

C

190

185

slide20

Flow Nets

El. = 200 ft

30 ft

El. = 200 ft

30 ft

A

Each stream tube carries

equal flow

F

B

El. = 170 ft

G

H

E

full

stream

tube

number of tubes (ns)= 2.2

number of drops (nd ) = 3 x 2 = 6

ns / nd = 2.2 / 6 = 0.37

H = 200 ft – 170 ft = 30 ft

full

stream

tube

195

D

C

partial

stream

tube

190

185

slide21

Flow Nets

El. = 200 ft

30 ft

El. = 200 ft

30 ft

A

Each stream tube carries

equal flow

F

B

El. = 170 ft

G

H

E

full

stream

tube

ns= 2.2

nd = 6

ns / nd= 2.2 / 6 = 0.37

Q = K grad h A

Q through square A = QA

QA = K [ (H / nd) / L ] (L * W)

QA = K H W / nd

L

L

195

D

C

A

L

partial

stream

tube

L

190

185

slide22

Flow Nets

El. = 200 ft

30 ft

El. = 200 ft

30 ft

A

Each stream tube carries

equal flow

F

3

B

2

1

El. = 170 ft

G

H

E

B

full

stream

tube

Since flow is steady state, flow

is continuous through the tube:

QA = QB = QC = Q

And since flow in all full tubes is equal:

QA = QB = QC =Q1 = Q2 = Q = K H W / nd

Total flow under dam is sum of all tubes:

ΣQ = Q1 + Q2 + Q3

195

D

C

A

partial

stream

tube

1

C

2

190

185

slide23

Flow Nets

El. = 200 ft

30 ft

El. = 200 ft

30 ft

A

Each stream tube carries

equal flow

F

3

B

2

1

El. = 170 ft

G

H

E

B

full

stream

tube

Total flow under dam is sum of all tubes:

ΣQ = Q1 + Q2 + Q3 = nf * Q

ΣQ = nf(K H W / nd)

ΣQ = K H W (nf/ nd)

195

D

C

A

partial

stream

tube

1

C

2

190

185

slide24

Flow Nets

El. = 200 ft

30 ft

El. = 200 ft

30 ft

A

Each stream tube carries

equal flow

F

3

B

2

1

El. = 170 ft

G

H

E

B

full

stream

tube

Total flow under this dam:

ΣQ = K H W (nf / nd )

ΣQ = K (30 ft) W (0.37)

If K = 100 ft/day and W = 1000 ft:

ΣQ = (100 ft/d) (30 ft) (1000 ft) (0.37)

ΣQ = 11.1 x 106 ft3/day,

About 11 million cubic feet per day

195

D

C

A

partial

stream

tube

1

C

2

190

185

fetter ch 45
Fetter, Ch. 4
  • 4.1 Introduction
  • 4.2Mechanical Energy
  • 4.3 Hydraulic Head
  • 4.4 Head in Water of Variable Density
  • 4.5 Force Potential and Hydraulic Head
  • 4.6 When is Darcy’s Law Applicable?
  • 4.7 Equations of Groundwater Flow
  • 4.8 Solutions of Flow Equations
  • 4.9 Gradient of Hydraulic Head
  • 4.10 Relationship of Flow to Gradient
  • 4.11 Flow Lines and Flow Nets
  • 4.12 Refraction of Flow Lines
  • 4.13 Steady Flow in Confined Aquifers
  • 4.14 Steady Flow in Unconfined Aquifers
slide26

Refraction of Flow Lines

Layer 1

K1 < K2

Layer 2

slide27

Refraction of Flow Lines

Continuity, or steady-state flow—demands that

Q1 = Q2

Note that:

a = b cosσ1

and

c = b cosσ2

K1 tan σ1

K2 tan σ2

=

fetter ch 46
Fetter, Ch. 4
  • 4.1 Introduction
  • 4.2 Mechanical Energy
  • 4.3 Hydraulic Head
  • 4.4 Head in Water of Variable Density
  • 4.5 Force Potential and Hydraulic Head
  • 4.6 When is Darcy’s Law Applicable?
  • 4.7 Equations of Groundwater Flow
  • 4.8 Solutions of Flow Equations
  • 4.9 Gradient of Hydraulic Head
  • 4.10 Relationship of Flow to Gradient
  • 4.11 Flow Lines and Flow Nets
  • 4.12 Refraction of Flow Lines
  • 4.13 Steady Flow in Confined Aquifers
  • 4.14 Steady Flow in Unconfined Aquifers
slide29

Steady Flow in a Confined Aquifer

Q = K (dh/dL) A

Q = K (dh/dL) b(1)

Q = K [(h1-h2)/L] b

QL/(Kb) = h1-h2

h2 = h1 - QL/(Kb)

Solve for h at any x:

hX= h1 - Qx/(Kb)

(Linear equation)

X

Q

Q

slide31

Steady Flow in an Unconfined Aquifer

Continuity, or steady-state flow—demands that

Q1 = Q2

At any x,

Q = K h dh/dL

But, h=h(x)

Thus, dh/dL = dh/dx

dh/dx= dh/dx (x)

Q

Q

slide32

Steady Flow in an Unconfined Aquifer

Dupuit Assumptions

  • Hydraulic gradient is equal to the slope of the water table.
  • Stream (flow) lines are horizontal.
  • Equipotential lines are vertical (follows from #2).
slide34

Steady Flow in an Unconfined Aquifer

Continuity, or steady-state flow—demands that

Q1 = Q2

At any x,

Q = K h dh/dL

But, h=h(x)

Thus, dh/dL = dh/dx

dh/dx= dh/dx (x)

Q

Q

slide35

Q = K h (dh/dx)

∫ Q dx= ∫ h dh

Q x = h2/2 + C

At x = 0, h = h1

At x = L, h = h2

Q (L – 0) =

(h22 – h12 )/2

QL = K (h22 – h12 )/2

h22 = h12- 2QL/K

(Dupuit Equation,

not linear )

Steady Flow in an Unconfined Aquifer

Q

Q

slide36

Steady Flow in an Unconfined Aquifer

h22 = h12- 2QL/K

(Dupuit Equation,

not linear )

Solve for h at any x:

hx2= h12 - 2Qx/K

Q = K (h12 - h22) / 2L

Q

Q

slide37

Boundary conditions?

Continuity?

Unconfined Flow with Infiltration or Evaporation

Q

slide38

Unconfined Flow with Infiltration or Evaporation

Q

Q = [K (h12 - h22) / 2L ] + w(L/2 – x)

Where is Q = 0?

slide39

Unconfined Flow with Infiltration or Evaporation

Q

Solve for h at any x:

hx2= h12– (h12 – h22) x/L + w/K(L –x) x

slide40

Unconfined Flow with Infiltration or Evaporation

Q

d =[L/2] –[K/w (h12 – h22)]/2L

slide41

Unconfined Flow with Infiltration or Evaporation

Q

If ratio K/w gets too big, then d <0

and no ground-water divide