Soil mechanics bk theme chapter 3 permeability

Department of Geotechnical Engineering 3.1 Darcys Law 7 For very low hydraulic gradients, the relationship between v and i is nonlinear Variation of discharge velocity with hydraulic gr

Chapter 3 Permeability of Soil

3.1 Darcys Law

3.2 Laboratory Determination of Hydraulic Conductivity

3.3 Permeability Test in the Field

1

3.4 Equivalent Hydraulic Conductivity in Stratified Soil

3.1 Darcys Law

2

Datum

vA2/2g

uA/ w

zA

h

z = 0

zA- elevation head

uA/w- pressure head

vA2 /2g - velocity head

h- total head

ℎ = 𝑧 + 𝑢

𝛾 +

𝑣 2𝑔

uA– water pressure

Bernoulli’s equation:

vA– velocity

w– unit weight of water

1

Department of Geotechnical Engineering

3.1 Darcys Law

3

Datum

uA/ w

zA

h

z = 0

zA- elevation head

hp = uA/w- pressure head

Total head:

ℎ = 𝑧 + 𝑢

𝛾

ℎ = 𝑧 + ℎ

or

The seepage velocity in soil is small

vA2 /2g  0

A 1

2

h1-h2

z1

Q

z2 Datum z = 0

w

u

2

Darcy’s Test

Flow rate (q) is linear with area cross section A and head loss on length L of soil sample

w

u

1

L

h h A k

q 1 2



L

h h k A

q

v  1 2

i k

v

L

h L

h h

i  1 2 

Velocity v

or with

k – hydraulic conductivity

i –hydraulic gradient

3.1 Darcys Law

3

4

3.1 Darcys Law

5

w

u



1

w

u



2

1

h = h1 - h2

z1

Q

z2

z = 0

Hydraulic gradient

𝑖 = ∆ℎ 𝐿

h – head loss between point 1 and point 2

L – length of flow from point 1

to point 2

∆ℎ = ℎ − ℎ = 𝑧 + 𝑢

𝑢 𝛾

3.1 Darcys Law

6

Darcy’s Law

v = k.i

v - discharge velocity, which is the quantity of water flowing in unit time through a unit gross cross-sectional area of soil at right angles to the direction of flow

i - hydraulic gradient

k - hydraulic conductivity (coefficient of permeability)

5

Department of Geotechnical Engineering

3.1 Darcys Law

7

For very low hydraulic gradients, the relationship between v and i

is nonlinear

Variation of discharge velocity with hydraulic gradient in clay

IN CLAY

3.1 Darcys Law

3 7

8

3.1 Darcys Law

Seepage velocity

v

s A v A v

n V

V L A

L A A

A v

v

v v

v

n

v

vs 

or or

9

3.2 Laboratory Determination of Hydraulic Conductivity

10

Q

h

L

Soil sample

t h A

L Q k







t i A

Q k





Collected data:

Q = volume of water collected

A = area of cross section of the soil specimen

L = length of the soil specimen

t = duration of water collection

h = difference of head between the inlet and outlet tank

Hydraulic Conductivity:

or

Constant-Head Test 9

Department of Geotechnical Engineering

3.2 Laboratory Determination of Hydraulic Conductivity

11

t A

Q



L h

 O

Constant-Head Test

Collected data:

Hydraulic Conductivity:

Volume of water (Q) is collected with change of the difference of head (h) between the inlet and outlet tank

Hydraulic conductivity determine from the chart relationship between Q/At and h/L

12

11

12

3.2 Laboratory Determination of Hydraulic Conductivity

13

Collected data:

a = area of cross section of the standpipe

A = area of cross section of the soil specimen

L = length of the soil specimen

t = 0 to t h = h0 to h

Hydraulic Conductivity:

Falling-Head Test

Q

L

h(t)

A

a















h

h t

L A

a

and

3.2 Laboratory Determination of Hydraulic Conductivity

14

Collected data:

Hydraulic Conductivity:

m = slope of line relationship between ln(h0/h) and t

Falling-Head Test











 h

h0 ln

1 m

Some variation of head h with time t

𝑘 = 𝑚 𝑎𝐿

𝐴

13

Department of Geotechnical Engineering

15

16

15

16

18

17

Department of Geotechnical Engineering

19

3.3 Permeability Test in the Field

20

q

Water table before pumping

Draw-down curve during pumping

r1

r2

Impermeable layer

dh

G.L

h

Test well

BH1, BH2 : Observation wells

Pumping test from a well in an unconfined permeable layer underlain by an impermeable stratum

i k A

q 

dr

dh rhk

q 2



1

2 2

1

2 h

h

r

r

dh h k r

dr

1

2 2

1 2

ln h h

r r q k







The rate of flow of groundwater into the well is equal to the rate

of discharge from pumping

Coefficient of permeability:

19

20

3.3 Permeability Test in the Field

21

Pumping test from a well penetrating the full depth in a confined aquifer The water can enter the test

well only from the aquifer of thickness H, the steady state of discharge is

Coefficient of permeability:

q

Draw-down curve during pumping

r1

r2

Impermeable layer

dh

G.L.

h

Tes

t well

Impermeable layer

BH1, BH2 : Observation wells

Water table before pumping

dr

dh k H r

q2



1

2 2

1

2

h

h r

r

dh k H r

dr

 

 22 11

2

ln h h H

r r q k







3.4 Equivalent Hydraulic Conductivity in Stratified Soil

22

Horizontal flow in stratified soil

Equivalent hydraulic conductivity:

21

Department of Geotechnical Engineering

3.4 Equivalent Hydraulic Conductivity in Stratified Soil

23

Vertical flow in stratified soil

Equivalent hydraulic conductivity:

24

23

24

26

25

Department of Geotechnical Engineering

27

27