Permeability Flow for b w

One-Dimensional Flow Through Soils

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Introduction

Why do we need to know about permeability and flow through soils?

Permeability

Soil is considered as a porous medium and the interconnected voids allow water to flow through it.

Water can flow from points of high energy to points of low energy.

Permeability

Soil is considered as a porous medium and the interconnected voids allow water to flow through it.

Water can flow from points of high energy to points of low energy.

The permeability k is used to quantify this property.

Permeability is the ease with which a liquid can flow through the pores of the soil

Ground Water

Ground Water Table

Hydrostatic condition when there is no flow

HeadHead: Head refers to the energy per unit of weight.

According to the Bernoulli’s equation, Total head = pressure head + velocity head + elevation head

where h = total headu = pressurev = velocityg = acceleration due to gravitygw = unit weight of water

Weight

EnergyHead =

Zg

vuh

w

++=2

2

g

Zg

vuh

w

++=2

2

g

Head

The elevation or potential head (Z) depends on the chosen datum.

The velocity of flow through soils is generally small (< 1cm/s) and we usually neglect the velocity head.

Thus >>

negligible

Zg

vuh

w

++=2

2

g

Zu

hw

+=g

P

z(P)

Datum

Definition of Head at a Point

(1)

Note

z is measured vertically upfrom the datum

)()(

)( PZPu

Phw

+=g

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

P

z P

thus

h P mw

w

( )

( )

=

= + =

1

41 5

wu w( ) = 4g

g

g

2. Calculation of head at X



2 m

5 mX

P

Impermeable stratum

1 m

1m

u X

z X

thus

h X m

w w

w

w

( )

( )

( )

=

=

= + =

g

g

g

4

4 5

The heads at P and X are identical. This implies that the head is constant throughout the region below a 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

( )

( )

( )

=

=

= + =

g

g

g

4

4 5

2 m

5 mX

P

Impermeable stratum

1 m

1m

3. Calculation of head at P

Choose datum at the water table


u P

z P

thus

h P m

w w

w

w

( )

( )

( )

=

= -

= - =

4

4

44 0

g

g

g

2 m

5 mX

P

Impermeable stratum

1 m

1m




u X

z X

thus

h X m

w w

w

w

( )

( )

( )

=

= -

= - =

1

1

1 0

g

g

g

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

2 m

5 mX

P

Impermeable stratum

1 m

1m




u X

z X

thus

h X m

w w

w

w

( )

( )

( )

=

= -

= - =

1

1

1 0

g

g

g

• 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 consider imaginary standpipes placed in the soil at the points where the head is required

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

Head

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

• proportional to the head difference h

• proportional to the cross-sectional area A

• inversely proportional to the length of sample L

Water flow through soil

h

L

Soil Sample

Darcy’s Law

q kAh

L=

(2a)Thus

where k is the coefficient of permeability or hydraulic conductivity.

Equation (2a) may be written asq = kAi

orv = k i (2b)

where i = h/L the hydraulic gradient

v = q/A the Darcy or superficial velocity

k the coefficient of permeability

q = quantity of water flowing through soil per unit time => units (volume/time) معدل السریان

Q = quantity of flow through soil

السائل المتدفق ) حجم(كمیة

Q = qt => units (volume)

Coefficient of Permeability (k)(Factors influence)العوامل المؤثرة على معامل النفاذیة

k = (gw/w) k(factors)

gw = unit weight of water or fluidw= viscosity of water or fluid (note temperature)

k(factors) Porosity (Void ratio)Pore size (Particle size)Tortuosity (particle shape)Saturation

عوامل متعلقة بالتربة

عوامل متعلقة بالمائع الذي

یسري

Manometers(Peizometers)

L

inlet

outlet

H

constant headdevice

device for flowmeasurement

load

porous disk

Constant Head Permeameter

sample

Measurement of permeability

The volume discharge Q during a suitable time interval t is collected.

The difference in head H over a length L is measured by means of manometers.

Knowing the cross-sectional area A, Darcy’s law gives

It can be seen that in a constant head permeameter::

(3)

Constant head permeameter

L

HkA

t

Q=

AHt

QLk =

H2

H1H

L

Falling Head Permeameter

Standpipe ofcross-sectionalarea a

Sample of area A

porous disk

Measurement of permeability

H2

H1

H

L

Standpipeof area

a

Sample of area

A

Initially H=H1 at time t=t1Finally H=H2 at time t=t2.

kaL

A

n H H

t t

=

-

l ( / )1 2

2 1

Falling head permeameter

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 Clays

Fissured & Weathered Clays

Typical Permeability Ranges (metres/second)

Typical Coefficient of permeability values

Soils exhibit a wide range of permeabilities and while particle size may vary by about 3-4 orders of magnitude permeability may vary by about 10 orders of magnitude.

Coefficient of Permeability (k)Order of Magnitude

k(cm/sec)

Clean Gravel 102 – 1

Clean Course to medium Sand 1 – 10-2

Fine Sand 10-2 – 10-4

Silt (Mixture of Sand & Silt) 10-4 – 10-6

Clay 10-6 – 10-12

(Typical 10-7)

Empirical relationships for k

• For a homogeneous soil, coefficient of permeability k depends on the soil fabric or structural arrangement of the soil grains.

• A number of empirical relationships have been proposed linking k to void ratio and grain size for coarse-grained soils:

Hazen (1930)• where C = a constant that is generally varies between

0.5 and 1.5. For coarse and fine sand, C 1.• D10 = the effective particle size, in mm• (for clean sands with less than 5% passing the No.200 sieve)

210sec)/( CDcmk =

Effect of Temperature on Permeability (k)

k is a function of viscosity and the unit weight of the pore fluid,

g wk

The viscosity of fluid, such as water, varies with temperature. Hence, k also changes with different temperatures.

It is conventional to express the value of k at a temperature of 20 C under the assumption that gw does not vary too much with temperature,

CTC20

CT

C20kk

=

Effect of Temperature on Permeability (k)

Variation of (water)C20CT

C20CT C20CT T (C) T (C)

15 1.135 23 0.931

16 1.106 24 0.910

17 1.077 25 0.889

18 1.051 26 0.869

19 1.025 27 0.850

20 1.000 28 0.832

21 0.976 29 0.814

22 0.953 30 0.797

Seepage Force

Seepage force per unit volume ww

s iL

hj g=

g=

Stratified Flow

Stratified Flow

Permeability Flow for b w

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