4. Soil Permeability and
Seepage
Soil Mechanics
2010 - 2011
Permeability
‡
Is a measure of how easily a fluid (water) can pass through
a porous medium (soil).
Loose Soil
High permeability
Dense Soil
Low permeability
Dr. Manal A. Salem – Soil Mechanics
1
Soil Permeability
‡
Applications (examples):
1.
Water wells
a.
b.
2.
Earth dams
a.
b.
3.
Water production
Dewatering
Estimate quantity of water seeping through the dam
Evaluating stability of dam
Ground improvement by preloading
Dr. Manal A. Salem – Soil Mechanics
Darcy’s Law
‡
‡
‡
‡
Water moves through soil with
discharge Q, and velocity v.
Q = A.v
v∝ i
v ∝ h/L
v = kh/L
Q = Akh/L
where:
„
„
„
„
„
„
„
h
1
Q = V/t
Soil
2
Q = water discharge (volume/time)
A = area perpendicular to flow direction
v = velocity of flow (length/time)
L
i = hydraulic gradient = h/L
h = total head difference
Direction of flow
L = length parallel to flow direction
k = coefficient of soil permeability
A
Dr. Manal A. Salem – Soil Mechanics
2
Coefficient of Permeability “k”
‡
Also called hydraulic conductivity
‡
k=v/i
‡
Define k: “the velocity of water flowing
through a soil medium under a unit hydraulic
gradient”
‡
Note:
flow of water through soil is governed by:
1.
2.
Head difference (i=h/l)
Soil permeability (k)
Dr. Manal A. Salem – Soil Mechanics
Bernoulli’s Equation
G.S.
hv1=v12/2g
Clay
hp1=u1/γw
1
ht1
Water
flow
Sand
z1
Clay
‡
h t = z + hp + h v
Δh
hv2=v22/2g
hp2=u2/γw
2
ht2
z2
Datum
where:
z = position head (elevation head)
hp = pressure head = u/γw: u = pore-water pressure
hv = velocity head = v2/2g
very small in soil and may
be neglected
Dr. Manal A. Salem – Soil Mechanics
3
Bernoulli’s Equation
G.S.
hv1=v12/2g
Clay
hp1=u1/γw
1
ht1
Water
flow
Sand
Δh
2/2g
hv2=v2
hp2=u2/γw
z1
2
ht2
z2
Clay
Datum
‡
‡
‡
h t ~ z + hp
hp is determined using piezometer (later)
Δh = total head difference, if Δh = 0, no flow.
Dr. Manal A. Salem – Soil Mechanics
Seepage and Discharge Velocities
‡
Discharge velocity (v):
„
„
„
‡
velocity of flow through entire cross-section.
Q = Av
Can be measured.
Seepage velocity (vs):
„
„
„
velocity of flow through voids.
Q = Avvs
Can’t be measured, only calculated, how?
Total area
(A)
L
Area of voids
(Av)
Dr. Manal A. Salem – Soil Mechanics
4
Seepage and Discharge Velocities
‡
Q = Av = Avvs
Therefore: vs = v ( A/AV)
Multiplying both areas (A and Av) by the length of the
medium (L)
vS = v ( AL / AVL ) = v ( VT / VV )
where:
VT = total volume of sample
VV = volume of voids within sample
By Definition, Vv / VT = n, the soil porosity
Thus
vS = v/ n
Dr. Manal A. Salem – Soil Mechanics
Factors affecting “k”
‡
Soil type
ksand > kclay
‡
Void ratio
kloose sand > kdense
‡
A
sand
Particles orientation
kB > kA
‡
Soil Structure
kflocculated > kdispersed
‡
Type of fluid
Viscosity
,k
‡
B
flocculated
A
dispersed
Temperature
Temperature
, Viscosity
,k
Dr. Manal A. Salem – Soil Mechanics
5
Laboratory determination of “k”
1.
Constant head test
2.
Falling head test
Dr. Manal A. Salem – Soil Mechanics
Constant Head Test
‡
Head difference
constant
‡
Apply Darcy’s law:
Continuous
water supply
c
Q = Av
V/t = Akh/L
k = VL/Aht
Overflow:
Volume V in
Time t
where
V = volume of water collected
in time = t
h = constant head difference
Direction
of flow
A = x-sectional area of soil
specimen
L = length of soil specimen
‡
Suitable for coarse-grained soils.
Dr. Manal A. Salem – Soil Mechanics
6
Falling Head Test
‡
Head is variable
‡
Coefficient of permeability (k) can
be calculated using the following
relationship:
a
h
Ak
ln 1 =
T
h2 La
T
ho
hf
where:
Overflow
h1 = initial head difference at time = 0
h2 = final head difference at time T
a = x-sectional area of standpipe
A = x-sectional area of soil specimen
Direction of
flow
A
L = length of soil specimen
‡
Suitable for fine-grained soils.
Dr. Manal A. Salem – Soil Mechanics
Falling Head Test
‰ Q = A v = A k i = A.k.
‰ Q at time dt =
‰ From 1 and 2:
−
dh.a
dt
A.k.
aL dh
.
Ak h
h
T
aL f dh
‰ ∫ dt = −
Ak h∫o h
0
hf
aL
ln h ho
‰ T =−
Ak
h o Ak
‰ ln h = La T
f
‰ dt =
−
[
]
h
-------- (1)
L
-------- (2)
h
dh.a
= −
L
dt
a
T
ho
hf
Overflow
Direction of
flow
A
Dr. Manal A. Salem – Soil Mechanics
7
Limitations of permeability lab tests
‡
Non-homogeneity of soil
‡
Anisotropy of soil
‡
Sampling disturbance
‡
Cracks and inclusions
Dr. Manal A. Salem – Soil Mechanics
Field determination of “k”
‡
Definitions:
„
„
Aquifer: a water-bearing layer of soil with considerable
amount of water.
Confined versus unconfined aquifers.
Dr. Manal A. Salem – Soil Mechanics
8
Field determination of “k”
‡
Definitions:
„
Piezometer: a small-diameter pipe used to measure the
groundwater head in aquifers.
Piezometers
Dr. Manal A. Salem – Soil Mechanics
Field determination of “k”
1.
Gravity (unconfined) Aquifer:
Initial water table
H
Aquitard
Dr. Manal A. Salem – Soil Mechanics
9
Field determination of “k”
1.
Gravity (unconfined) Aquifer:
Pumping well
Initial water table
H
Aquitard
Dr. Manal A. Salem – Soil Mechanics
Field determination of “k”
1.
Gravity (unconfined) Aquifer:
Pumping well
Q
Initial water table
Draw down water table
H
Aquitard
Dr. Manal A. Salem – Soil Mechanics
10
Field determination of “k”
1.
Gravity (unconfined) Aquifer:
Pumping well
Piezometer (1)
Q
r1
r2
Piezometer (2)
Initial water table
∆h2
∆h1
Draw down water table
H
h2
h1
Aquitard
Dr. Manal A. Salem – Soil Mechanics
Field determination of “k”
1.
Gravity (unconfined) Aquifer:
•Pump water from
well at a constant rate
(Q) until reach steady
state (water level in
observation wells is
constant)
•Field measurements:
Q, r1, r2, h1, h2,
where:
h1 = H – Δh1
h2 = H – Δh2
Pumping well
Initial water
table
Q
Piezometer (1)
r2
r1
Piezometer (2)
∆h2
∆h1
Draw down water
table
h1
h2
H
Aquitard
•Calculate k
k=
Q ln(r2 / r1 )
π h22 − h12
Dr. Manal A. Salem – Soil Mechanics
11
Field determination of “k”
2.
Artesian (confined) Aquifer:
Aquitard
D
Aquitard
Dr. Manal A. Salem – Soil Mechanics
Field determination of “k”
2.
Artesian (confined) Aquifer:
Initial piezometric
surface
Pumping well
D
Aquitard
Dr. Manal A. Salem – Soil Mechanics
12
Field determination of “k”
2.
Artesian (confined) Aquifer:
Initial piezometric
surface
Q
Pumping well
Draw down
piezometric line
H
D
Aquitard
Dr. Manal A. Salem – Soil Mechanics
Field determination of “k”
2.
Artesian (confined) Aquifer:
Initial piezometric
surface
Q
Pumping well
r1
Piezometer (1)
r2
Piezometer (2)
∆h2
∆h1
Draw down
piezometric line
h1
h2
H
D
Aquitard
Dr. Manal A. Salem – Soil Mechanics
13
Field determination of “k”
2.
Artesian (confined) Aquifer:
•Pump water
from well at a
constant rate (Q)
until reach
steady state
Initial piezometric
Pumping well
surface
•Field
measurements:
Q, r1, r2, h1, h2,
where:
h1 = H – Δh1
h2 = H – Δh2
Q
Piezometer (1)
r2
r1
Piezometer (2)
∆h2
∆h1
Draw down
piezometric line
h1
h2
H D
Aquitard
•Calculate k
k=
Q ln(r2 / r1 )
2πD h2 − h1
Dr. Manal A. Salem – Soil Mechanics
Field determination of “k”
‡
Overcomes the limitations of laboratory tests.
‡
Much more expensive compared to laboratory
tests.
Dr. Manal A. Salem – Soil Mechanics
14
Empirical Correlations for “k”
1.
Coarse-grained soils
„
Hazen’s (1930):
k (cm / sec) = cD102
where
c = constant ranging from 1 to 2
D10 = effective grain size in mm
„
Chapuis (2004):
⎡
e3 ⎤
k (cm / sec) = 2.4622⎢ D102
⎥
(1 + e) ⎦
⎣
where
0.7825
e = void ratio
D10 = effective grain size in mm
Dr. Manal A. Salem – Soil Mechanics
Empirical Correlations for “k”
2.
Fine-grained soils
⎛ en ⎞
⎟⎟
k = C ⎜⎜
⎝1+ e ⎠
where
C and n = constants determined experimentally
e = void ratio
Dr. Manal A. Salem – Soil Mechanics
15
Empirical Correlations for “k”
Example: A clayey soil was tested in the lab and the following
values were determined:
Void ratio
k (cm/sec)
1.1
0.302 x 10-7
0.9
0.12 x 10-7
Estimate “k” for void ratio = 0.75
‰ Answer:
⎛ (1.1) n ⎞
⎟⎟
0.302x10 = C⎜⎜
1
1
.
1
+
⎠
⎝
−7
⎛ en ⎞
⎟⎟
k = C⎜⎜
1
+
e
⎠
⎝
⎛ (0.9) n ⎞
⎟⎟
0.120x10−7 = C⎜⎜
⎝ 1 + 0.9 ⎠
Dr. Manal A. Salem – Soil Mechanics
Empirical Correlations for “k”
Example: A clayey soil was tested in the lab and the following
values were determined:
Void ratio
k (cm/sec)
1.1
0.302 x 10-7
0.9
0.12 x 10-7
Estimate “k” for void ratio = 0.75
‰ Answer:
0.302x10−7
0.120x10−7
⎛ (1.1) n ⎞
⎜⎜
⎟⎟
1
1
.
1
+
⎠
=⎝
⎛ (0.9) n ⎞
⎜⎜
⎟⎟
⎝ 1 + 0.9 ⎠
n = 5.098
Dr. Manal A. Salem – Soil Mechanics
16
Empirical Correlations for “k”
Example: A clayey soil was tested in the lab and the following
values were determined:
Void ratio
k (cm/sec)
1.1
0.302 x 10-7
0.9
0.12 x 10-7
Estimate “k” for void ratio = 0.75
‰ Answer:
⎛ (1.1)5.098 ⎞
⎟⎟
0.302x10−7 = C⎜⎜
⎝ 1 + 1.1 ⎠
C = 0.390x10−7
⎛ (0.75)5.098 ⎞
⎟⎟ = 0.051x10−7 cm / sec
k = 0.390x10 ⎜⎜
⎝ 1 + 0.75 ⎠
−7
Dr. Manal A. Salem – Soil Mechanics
Typical Values of “k”
Soil Type
k (cm/sec)
Gravel
100 – 10-1
Coarse Sand
10-1 – 10-2
Fine Sand
10-2 – 10-3
Silty Sand
10-3 – 10-4
Silt
10-4 – 10-5
Clay
<10-6
Dr. Manal A. Salem – Soil Mechanics
17