Soil Mechanics - Wikispacesustce424soilmechanics01.wikispaces.com/file/view/10_Permeability... · This test applies a constant head of water to each end of a soil in a “permeameter”

X. PERMEABILITY Soil Mechanics

HYDRAULIC CONDUCTIVITY One-dimensional flow

2

Large Earth Dam

SHELL

FOUNDATION

SHELL

CORE

blanket

filter

cutoff

crest

riprap

free board

What is permeability? a measure of how easily a fluid (e.g., water)

can pass through a porous medium (e.g., soils).

Loose soil Dense soil

water

- high permeability

- easy to flow

- low permeability

- difficult to flow

Importance of permeability Permeability influences the rate of

settlement of a saturated soil under load.

The design of earth dams is very much based upon the permeability of the soils

used.

The stability of slopes and retaining

structures can be greatly affected by the

permeability of the soils involved.

Filters made of soils are designed based

upon their permeability.

The study of permeability is important for: Estimating the quantity of underground

seepage.

Investigating problems involving pumping

seepage of water for underground

constructions.

Analyzing the stability of earth dams and earth retaining walls subjected to

seepage forces.

Reynold’s Number a dimensionless number which indicates the

characteristic state of fluid motion.

Bernoulli’s Equation

1. Kinetic energy

datum

z

fluid particle

The energy of a fluid particle is made of:

2. Pressure energy

3. Elevation energy

- due to velocity

- due to pressure

- due to elevation (z) with respect to a datum

Total head =

datum

z

fluid particle

Expressing energy in unit of length:

Velocity head

+

Pressure head

+

Elevation head


Total head =

datum

z

fluid particle

For flow through soils, velocity (and thus

velocity head) is very small. Therefore,

Velocity head

+

Pressure head

+

Elevation head

0


If flow is from A to B, total head is

higher at A than at B.

water

A B

Energy is dissipated

in overcoming the

soil resistance and

hence is the head

loss.



Some Notes

Reynold’s Number a dimensionless number which indicates the

characteristic state of fluid motion.

Some Notes

Pressure head = pore water pressure/w

Elevation head = height above the

selected datum

At any point within the flow regime:

Some Notes

Hydraulic gradient (i) between A

and B is the total head loss per

unit length.

water

A B

AB

BA

L

hhi

length AB, along

the stream line

Henri Darcy in 1856 derived an empirical formula for the behavior of flow through saturated soils. He found that the quantity of water (q) per sec

flowing through a cross-sectional area (A) of soil under hydraulic gradient (i)

can be expressed by the formula:

where, v: discharge velocity, which is the quantity of water

flowing in unit time through a unit gross cross-sectional area of soil (cm/s). k: coefficient of permeability or hydraulic conductivity

(cm/s). q: flow rate (cm3/s). Q: volume of collected water (cm3). A: cross-sectional area (cm3). i: hydraulic gradient.

Darcy’s Law

Some Notes

Seepage velocity, vs: is the

actual velocity of water

through the void spaces. vs is greater then v.

Hydraulic Conductivity

Factors: fluid viscosity, pore-size

distribution, grain-size distribution,

void ratio, roughness of mineral

particles, and degree of soil

saturation

Hydraulic Conductivity

𝒌𝑻𝟏

𝒌𝑻𝟐

=𝜼𝑻𝟐

𝜼𝑻𝟐

𝜸𝒘𝑻𝟏

𝜸𝒘𝑻𝟐

𝒌𝟐𝟎℃ =𝜼𝑻℃

𝜼𝟐𝟎℃𝒌𝑻℃

X.1. PERMEABILITY Soil Mechanics

Determination of Hydraulic Conductivity

Constant Head Test The constant head test is used primarily for coarse-grained soils.

It is based on the assumption of laminar flow where k is independent of i (low values of i).

This test applies a constant head of water to each end of a soil in a “permeameter” (ASTM D2434).

After a constant flow rate is established, water is collected in a graduated flask for a known duration.

Constant Head Test

Laboratory Test

Falling Head Test The falling head test is used for both coarse-grained soils as well as fine-grained soils.

Same procedure in constant head test except:

Record initial head difference, h1 at t = 0

Allow water to flow through the soil specimen

Record the final head difference, h2 at time t = t2

Collect water at the outlet, Q (in ml) at time t ≈ 60 sec

Falling Head Test

Laboratory Test

Pumping Test

Field Test

Pumping Test

Field Test

X.2. PERMEABILITY Soil Mechanics Empirical Relations for Hydraulic Conductivity

Empirical Relations for Hydraulic Conductivity

Hazen (1930)

Kozeny-Carman Equation (Kozeny, 1927; Carman, 1938, 1956)

Granular Soils


Granular Soils


Carrier (2003) Granular Soils


Cohesive Soils Tavenas et al. (1983) Samarasinghe et al. (1982)


X.3. PERMEABILITY Soil Mechanics

Equivalent Hydraulic Conductivity

in Stratified Soil

For parallel direction of flow The equivalent hydraulic conductivity in the horizontal direction (kH(eq)) is:

For parallel direction of flow

𝑞 = 𝑣. 1. 𝐻

𝑞 = 𝑣1. 1. 𝐻1 + 𝑣2. 1. 𝐻2 + 𝑣3. 1. 𝐻3 + ⋯ + 𝑣𝑛. 1. 𝐻𝑛

𝑣 = 𝑘𝐻(𝑒𝑞). 𝑖𝑒𝑞; 𝑣1 = 𝑘𝐻1. 𝑖1; 𝑣2 = 𝑘𝐻2. 𝑖2; 𝑣3 = 𝑘𝐻3. 𝑖3; … ; 𝑣𝑛 = 𝑘𝐻𝑛. 𝑖𝑛;

𝑖𝑒𝑞 = 𝑖1 = 𝑖2 = 𝑖3 = ⋯ = 𝑖𝑛

𝒌𝑯(𝒆𝒒) =𝟏

𝑯𝒌𝑯𝟏. 𝑯𝟏 + 𝒌𝑯𝟐. 𝑯𝟐 + 𝒌𝑯𝟑. 𝑯𝟑 + ⋯ + 𝒌𝑯𝒏. 𝑯𝒏

For perpendicular direction of flow The equivalent hydraulic conductivity in the vertical direction (kV(eq)) is:

For parallel direction of flow

𝑣 = 𝑣1 = 𝑣2 = 𝑣3 = ⋯ = 𝑣𝑛

ℎ = ℎ1 + ℎ2 + ℎ3 + ⋯ +ℎ𝑛

𝑘𝑣(𝑒𝑞).ℎ

𝐻= 𝑘𝑣1. 𝑖1 = 𝑘𝑣2. 𝑖2 = 𝑘𝑣3. 𝑖3 = ⋯ = 𝑘𝑣𝑛. 𝑖𝑛;

ℎ = 𝐻1. 𝑖1 + 𝐻2. 𝑖2 + 𝐻3. 𝑖3 + ⋯ + 𝐻𝑛. 𝑖𝑛

𝒌𝒗(𝒆𝒒) =𝑯

𝑯𝟏𝒌𝒗𝟏

+𝑯𝟐𝒌𝒗𝟐

+𝑯𝟑𝒌𝒗𝟑

+ ⋯ +𝑯𝒏𝒌𝒗𝒏

Problem 1 For a constant head laboratory permeability test on a fine sand, the

following values are given:

Length of specimen = 250 mm

Diameter of specimen = 64 mm

Head difference = 460 mm

Water collected in 2 min = 0.51 cm3

Void ratio of the soil specimen = 0.46

Temperature of water = 24°C

Determine:

1.1 Hydraulic conductivity of the soil in cm/min (4.308x10-3 cm/min)

1.2 Discharge velocity in cm/min (7.927x10-3 cm/min)

1.3 Seepage velocity in cm/min (0.0252 cm/min)

1.4 Hydraulic conductivity for the soil at 20°C (3.920x10-3 cm/min)

Problem Set 7

Problem 2 For a constant head permeability test,

the following values are given:

Length of specimen = 300 mm

Area of specimen = 32 cm2

k = 0.0244 cm/sec

The head difference was slowly

changed in steps to 800, 700, 600, 500, and 400 mm. Calculate and plot the

rate of flow, q, through the specimen, in

cm3/sec, against the head difference.

Problem Set 7

Problem 3 For a falling head permeability test, the following values are given:

Length of specimen = 38 cm

Area of specimen = 19 cm2

k = 0.175 cm/min

What should be the area of the standpipe for the head to drop from 64 cm to 30

cm in 8 min? (0.924 cm2)

Problem Set 7

Problem 4 An unconfined aquifer is known

to be 32 m thick below the

water table. A constant discharge of 2 m3/min is

pumped out of the aquifer

through a tube well till the

water level in the tube well

becomes steady. Two

observation wells at distances of 15 m and 70 m from the tube

well show falls of 3 m and 0.7 m

respectively from their static water levels. Find the permeability

of the aquifer. (0.0118 cm/sec)

Problem Set 7

Problem 5 The hydraulic conductivity of a clayey soil is 3x10-7 cm/sec. The

viscosity of water at 25°C is 0.0911x10-4 g.sec/cm2. Calculate the

absolute permeability of the soil. (2.733x10-12 cm2)

Problem Set 7

Problem 6 A permeable soil layer is underlain

by an impervious layer, as shown in the figure. With k = 4.8x10-3

cm/sec for the permeable layer,

calculate the rate of seepage through it

in m3/hr/m width if H=3 m and α = 50.

(0.045 m3/hr/m)

Problem Set 7

Problem 7 Three layers of soil is shown with the corresponding values of

coefficient of permeability. Determine the following:

7.1 Equivalent horizontal coefficient of permeability (5.167x10-3 cm/sec)

7.2 Equivalent vertical coefficient of permeability (5.035x10-3 cm/sec)

7.3 Ratio of equivalent coefficient of permeability (1.026)

3 m

4 m

5 m

k = 4x10-3 cm/sec

k = 5x10-3 cm/sec

k = 6x10-3 cm/sec

Problem Set 7

Problem 8 For the figure shown, determine the total rate of flow in cm3/sec. (0.667 cm3/sec)

Problem Set 7

Tube

(100 mm x 100 mm)

Constant head

difference = 300 mm

DATUM

Soil k (cm/sec)

A 10-2

B 3x10-3

C 4.9x10-4


.

Problem Set 7

Soil k (cm/sec)

A 10-2

B 3x10-3

C 4.9x10-4

Tube

(100 mm x 100 mm)

Constant head

difference = 300 mm

DATUM


.

Problem Set 7

Soil k (cm/sec)

A 10-2

B 3x10-3

C 4.9x10-4 Constant head

difference = 300 mm

DATUM

Tube

(100 mm x 100 mm)

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Soil Mechanics - Wikispacesustce424soilmechanics01.wikispaces.com/file/view/10_Permeability... · This test applies a constant head of water to each end of a soil in a “permeameter”