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2

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A measure of how easily a fluid (e.g., water)can pass through a porous medium (e.g.,soils)

3

Loose soil

- easy to flow

- high permeability

Dense soil

- difficult to flow

- low permeability

water

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1. Kinetic energy

4

datum

z

fluid particle

The energy of a fluid particle ismade of:

2. Strain energy

3. Potential energy

- due to velocity

- due to pressure

- due to elevation (z) with respect to adatum

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Total head =

5

datum

z

fluid particle

Expressing energy in unit of length:

Velocity head

+Pressure head

+

Elevation head

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Total head =

6

datum

z

fluid particle

For flow through soils, velocity (and thus

velocity head) is very small. Therefore,

Velocity head

+Pressure head

+

Elevation head

0

Total head = Pressure head + Elevationhead

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If flow is from A to B, total head is higher at Athan at B.

7

water

A B

Energy is dissipatedin overcoming thesoil resistance andhence is the headloss.

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Pressure head = pore water pressure/ w

8

Elevation head = height above the selecteddatum

At any point within the flow regime:

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Hydraulic gradient (i) between A and B is thetotal head loss per unit length.

9

water

A B

AB

B A

l

TH TH i

length AB, along thestream line

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In laminar flow each particle travels along adefinite path which never crosses the path ofother particles

In turbulent flow the paths are irregular andtwisting, crossing and recrossing at random.

Since pores of most soils are small, flow throughthem is invariably laminar

10

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Flow through a Dam

UnsaturatedSoil

Flow of water

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P

z(P)

Datum

Definition of Head at a Point

h P u P

z Pw

w( )

( )( )

(1)

Note

z is measured vertically upfrom the datum

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1. Calculation of head at P

Choose datum at the top of the impermeable layer

2 m

5 mX

P

Impermeable stratum

1 m

1m

Example: Static water table

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1. Calculation of head at P

Choose datum at the top of the impermeable layer

2 m

5 mX

P

Impermeable stratum

1 m

1m

Example: Static water table

Pwu w( ) 4

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1. Calculation of head at P

Choose datum at the top of the impermeable layer

2 m

5 mX

P

Impermeable stratum

1 m

1m

Example: Static water table

Pz P( ) 1wu w( ) 4

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1. Calculation of head at P

Choose datum at the top of the impermeable layer

2 m

5 mX

P

Impermeable stratum

1 m

1m

Example: Static water table

Pz P

thus

h P mww

( )

( )

1

41 5

wu w( ) 4

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2. Calculation of head at X

Choose datum at the top of the impermeable layer

Example: Static water table

2 m

5 mX

P

Impermeable stratum

1 m

1m

u Xw w( )

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2. Calculation of head at X

Choose datum at the top of the impermeable layer

Example: Static water table

2 m

5 mX

P

Impermeable stratum

1 m

1m

u X

z Xw w( )

( ) 4

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2. Calculation of head at X

Choose datum at the top of the impermeable layer

Example: Static water table

2 m

5 mX

P

Impermeable stratum

1 m

1m

u X

z X

thus

h X m

w w

w

w

( )

( )

( )

4

4 5

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The heads at P and X are identical does this imply that the headis constant throughout the region below a static water table?

2. Calculation of head at X

Choose datum at the top of the impermeable layer

Example: Static water table

2 m

5 mX

P

Impermeable stratum

1 m

1m

u X

z X

thus

h X m

w w

w

w

( )

( )

( )

4

4 5

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2 m

5 mX

P

Impermeable stratum

1 m

1m

3. Calculation of head at P

Choose datum at the water table

Example: Static water table

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2 m

5 mX

P

Impermeable stratum

1 m

1m

3. Calculation of head at P

Choose datum at the water table

Example: Static water table

u Pw w

( ) = 4

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2 m

5 mX

P

Impermeable stratum

1 m

1m

3. Calculation of head at P

Choose datum at the water table

Example: Static water table

u P

z Pw w

( )

( )

=

= -

4

4

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2 m

5 mX

P

Impermeable stratum

1 m

1m

3. Calculation of head at P

Choose datum at the water table

Example: Static water table

u P

z P

thus

h P m

w w

w

w

( )

( )

( )

=

= -

= - =

4

4

44 0

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2 m

5 mX

P

Impermeable stratum1 m

1m

4. Calculation of head at X

Choose datum at the water table

Example: Static water table

u Xw w( ) 1

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2 m

5 mX

P

Impermeable stratum1 m

1m

4. Calculation of head at X

Choose datum at the water table

Example: Static water table

u X

z X

w w( )

( )

1

1

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2 m

5 mX

P

Impermeable stratum1 m

1m

4. Calculation of head at X

Choose datum at the water table

Example: Static water table

u X

z Xthus

h X m

w w

w

w

( )

( )

( )

1

1

1 0

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Again, the head at P and X is identical, but the value is different

2 m

5 mX

P

Impermeable stratum1 m

1m

4. Calculation of head at X

Choose datum at the water table

Example: Static water table

u X

z Xthus

h X m

w w

w

w

( )

( )

( )

1

1

1 0

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The value of the head depends on the choice of datum

Head

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The value of the head depends on the choice of datum

Differences in head are required for flow (not pressure)

Head

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The value of the head depends on the choice of datum

Differences in head are required for flow (not pressure)

2 m

5 mX

P

Impermeable stratum

1 m

1m

It can be helpful to considerimaginary standpipesplaced in the soil at thepoints where the head isrequired

Head

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The value of the head depends on the choice of datum

Differences in head are required for flow (not pressure)

2 m

5 mX

P

Impermeable stratum

1 m

1m

It can be helpful to considerimaginary standpipesplaced in the soil at thepoints where the head isrequired

The head is the elevation of the water level in thestandpipe above the datum

Head

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Darcy found that the flow (volume per unit time) was

proportional to the head difference Dh

proportional to the cross-sectional area A

inversely proportional to the length of sample DL

Water flow through soil

h

DL

Soil Sample

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Darcys Law

Q kA hL

(2a)Thus

where k is the coefficient of permeability or hydraulicconductivity.

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Darcys Law

Q kA hL

(2a)Thus

where k is the coefficient of permeability or hydraulicconductivity.

Equation (2a) may be written as

Q k Ai

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Darcys Law

Q kA hL

(2a)Thus

where k is the coefficient of permeability or hydraulicconductivity.

Equation (2a) may be written as

or

Q k Ai

v = k i (2b)

where i = Dh/DL the hydraulic gradient

v = Q/A the Darcy or superficial velocity

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40

applicable to relativelypervious or coarse grainedsoils

From Darcys law

M t f bilit

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H2

H1H

L

Fig. 5 Falling Head Permeameter

Standpipe ofcross-sectionalarea a

Sampleof area A

porous disk

Measurement of permeability

F lli h d t

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Analysis

H2

H1H

L

Standpipeof area

a

Sampleof area

A

Consider a time interval dt

The flow in the standpipe =

Falling head permeameter

a H

t

d

d

F lli g h d t

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Analysis

H2

H1H

L

Standpipeof area

a

Sampleof area

A

Consider a time interval dt

The flow in the standpipe =

The flow in the sample =

Falling head permeameter

a H

t

d

d

k A HL

F lli g h d t

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Analysis

H2

H1H

L

Standpipeof area

a

Sampleof area

A

Consider a time interval dt

The flow in the standpipe =

The flow in the sample =

and thus

(4a)

Falling head permeameter

a H

t

d

d

k A HL

a dHdt

k A HL

Falling head permeameter

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H2

H1H

L

Standpipe

of areaa

Sample

of areaA

Solution

adHdt

kAHL

(4a)

Equation (4a) has the solution:

a n HkAL

tl ( ) (4b)

Falling head permeameter

Falling head permeameter

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H2

H1H

L

Standpipe

of areaa

Sample

of areaA

Solution

adHdt

kAHL

(4a)

Equation (4a) has the solution:

a n HkAL

tl ( ) (4b)

Initially H=H 1 at time t=t 1Finally H=H 2 at time t=t 2.

kaL

A

n H H

t t

l ( / )1 22 1

(4c)

Falling head permeameter

Typical permeability values

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10 -1 10 -2 10 -3 10 -4 10 -5 10 -6 10 -7 10 -8 10 -9 10 -10 10 -11 10 -12

Gravels Sands Silts Homogeneous ClaysFissured & Weathered Clays

Typical Permeability Ranges (metres/second)

Typical permeability values

Soils exhibit a wide range of permeabilities and while particle

size may vary by about 3-4 orders of magnitude permeabilitymay vary by about 10 orders of magnitude.

Definition of Hydraulic Gradients

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z

xx

z

A

B C

O

v k i

where

i h C h B

xand thus

v k h

x

x H x

x

x H

( ) ( )

For horizontal flow v=v x and k=k H and thus

(5a)

Definition of Hydraulic Gradients

Definition of Hydraulic Gradients

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For vertical flow v=v z and k=k V and thus

(5b)

v k i

where

i h A h B

zand thus

v k h

z

z V z

z

z V

( ) ( )

D

z

xx

z

A

B C

O

Definition of Hydraulic Gradients

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Valid for saturated soil samples underlaminar flow

Any flow with a Reynolds number less thanone is clearly laminar. Experimental tests have shown that flowswith Reynolds numbers up to 10 may still beDarcian, as in the case of groundwater flow.

52

Reynolds number

http://en.wikipedia.org/wiki/Reynolds_numberhttp://en.wikipedia.org/wiki/Reynolds_numberhttp://en.wikipedia.org/wiki/Reynolds_numberhttp://en.wikipedia.org/wiki/Reynolds_numberhttp://en.wikipedia.org/wiki/Reynolds_numberhttp://en.wikipedia.org/wiki/Reynolds_number

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Permeant Fluid PropertiesSoil Characteristics

1. Grain-size2. Void ratio3. Degree of saturation4. Presence of entrapped air and other foreign

matter.

53

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Dam

Impermeable bedrock

Fig. 8 Plane Flow under a Dam

Cross section of a long dam

(flow in the y direction is negligible)

SoilFlow

z

x

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A

B

C

D

x

z

v z

v xSoil

Element

Fig. 9 Flow intoa soil element

Net flow =(v x (B)-v x (D)) y z+(v z (C)-v z (A)) x y (6a)

For steady state seepage the net flow in will bezero, thus

v

x

v

z

x z 0

(6b)

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A

B

C

D

x

z

v z

v xSoil

Element

Fig. 9 Flow intoa soil element

Net flow =(v x (B)-v x (D)) y z+(v z (C)-v z (A)) x y (6a)

For steady state seepage the net flow in will bezero, thus

v

x

v

z

x z 0

(6b)

Continuity Equation

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y q v

xvz

x z 0Continuity Equation (6b)

Continuity Equation

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y q v

xvz

x z 0

Darcy's Law

v k h

x

v k hz

x H

z V

+

Continuity Equation

Darcys Law

+

(6b)

(5)

Continuity Equation

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vx

vz

x z 0

xk

hx z

k h

zH V( ) ( ) 0

Darcy's Law

v k h

x

v k hz

x H

z V

+

Continuity Equation

Darcys Law

Flow equation

+

(6b)

(7b)

(5)

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xk

hx z

k h

zH V( ) ( ) 0Flow equation (7b)

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xk

hx z

k h

zH V( ) ( ) 0Flow equation (7b)

k h

x k h

zH V

2

2

2

2 0 (7c)For a homogeneous soil

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xk

hx z

k h

zH V( ) ( ) 0Flow equation (7b)

k h

x k h

zH V

2

2

2

2 0 (7c)For a homogeneous soil

For an isotropic soil (7d 2

2

2

2 0h

x

h

z