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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
Bernoulli’s Equation
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
Bernoulli’s Equation
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.
Bernoulli’s Equation
Bernoulli’s Equation
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
Empirical Relations for Hydraulic Conductivity
Granular Soils
Empirical Relations for Hydraulic Conductivity
Carrier (2003) Granular Soils
Empirical Relations for Hydraulic Conductivity
Cohesive Soils Tavenas et al. (1983) Samarasinghe et al. (1982)
Empirical Relations for Hydraulic Conductivity
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 9 For the figure shown, determine the total rate of flow in cm3/sec. (0.300 cm3/sec)
.
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 10 For the figure shown, determine the total rate of flow in cm3/sec. (0.0809 cm3/sec)
.
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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