14.330 soil mechanics hydraulic conductivity &...
TRANSCRIPT
Slide 1 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
BERNOULLI’S EQUATION
Zg
vu hw
++=2
2
γWhere:
h = Total Head u = Pressure v = Velocity g = Acceleration due to Gravity γw = Unit Weight of Water
Slide 2 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
BERNOULLI’S EQUATION IN SOIL
Zg
vu hw
++=2
2
γ
Therefore:
Zu hw
+=γ
v ≈ 0 (i.e. velocity of water in soil is negligible).
v ≈ 0
Slide 3 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
Figure 5.1. Das FGE (2005).
CHANGE IN HEAD FROM POINTS
A & B (Δh) BA h hh −=Δ
⎟⎟⎠
⎞⎜⎜⎝
⎛+−⎟⎟
⎠
⎞⎜⎜⎝
⎛+=Δ B
w
BA
w
A ZuZu hγγ
BA h hh −=Δ
Lh i Δ
=
BA h hh −=Δ
Δh can be expressed in non-dimensional form
Where: i = Hydraulic Gradient L = Length of Flow between Points A & B
Slide 4 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
Figure 5.2. Das FGE (2005).
VELOCITY (v) vs. HYDRAULIC GRADIENT (i) General relationship shown in Figure 5.2
Three Zones: 1. Laminar Flow (I) 2. Transistion Flow (II) 3. Turbulent Flow (III) For most soils, flow is
laminar. Therefore:
v ∝ i
Slide 5 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
DARCY’S LAW (1856)
v = kiWhere: v = Discharge Velocity (i.e. quantity of water in unit time through unit cross-sectional area at right angles to the direction of flow) k = Hydraulic Conductivity (i.e. coefficient of permeability) i = Hydraulic Gradient
* Based on observations of flow of water through clean sands
Slide 6 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
Kk w
ηγ
=
HYDRAULIC CONDUCTIVITY (k)
Where: η = Viscosity of Water
K = Absolute Permeability (units of L2)
Soil Type k
(cm/sec) k
(ft/min) Clean Gravel 100-1 200-2
Coarse Sand 1-0.01 2-0.02
Fine Sand 0.01-0.001 0.02-0.002
Silty Clay 0.001-0.00001
0.002-0.00002
Clay < 0.000001 <0.000002
Typical Values of k per Soil Type
after Table 5.1. Das FGE (2005)
Slide 7 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
Figure 5.3. Das FGE (2005).
DISCHARGE & SEEPAGE VELOCITIES
svvAvAq ==Where: q = Flow Rate (quantity of water/unit time) A = Total Cross-sectional Area Av = Area of Voids vs = Seepage Velocity
Slide 8 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
DISCHARGE & SEEPAGE VELOCITIES
nv
eev
VVVV
vVVVvv
LAAAv
AAAvv
vAAAvq
s
v
s
v
v
svs
v
sv
v
svs
svsv
=⎟⎠
⎞⎜⎝
⎛ +=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+
=+
=
+=
+=
=+=
11
)(
)()()(
Figure 5.3. Das FGE (2005).
Slide 9 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
EXAMPLE PROBLEM
CL (Impervious Layer)
CL (Impervious Layer)
SM (k = 0.007 ft/min)
12 ft
175 ft
25 ft
15 ft
12°
GIVEN: REQUIRED: Find Hydraulic Gradient (i) and Flow Rate (q)
Slide 10 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
EXAMPLE PROBLEM – FIND i, q
CL (Impervious Layer)
CL (Impervious Layer)
SM (k = 0.007 ft/min)
12 ft
175 ft
25 ft
15 ft
Hydraulic Gradient (i):
067.0
12cos17512
=
⎟⎠
⎞⎜⎝
⎛=
Δ=
ftfti
Lhi
12°
ftftq
ftftftq
kiAq
min//109.6
)1)(12)(cos15)(067.0(min
007.0
33−×=
=
=
Rate of Flow per Time (q):
GIVEN: SOLUTION:
Slide 11 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
LABORATORY TESTING OF HYDRAULIC CONDUCTIVITY Constant Head (ASTM D2434)
Falling Head (no ASTM)
Figure 5.4. Das FGE (2005). Figure 5.5. Das FGE (2005).
Slide 12 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
LABORATORY TESTING OF HYDRAULIC CONDUCTIVITY
Constant Head (ASTM D2434)
AhtQLk =
tkiAAvtQ )(==Where: Q = Quantity of water collected over time t t = Duration of water collection
tLhkAQ ⎟⎠
⎞⎜⎝
⎛=
Figure 5.4. Das FGE (2005).
Slide 13 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
LABORATORY TESTING OF HYDRAULIC CONDUCTIVITY Falling Head (No ASTM)
2
110303.2hhLog
AtaLk =
dtdhaA
Lhkq −==
⎟⎠
⎞⎜⎝
⎛−=hdh
AkaLdt
Where: A = Cross-sectional area of Soil a = Cross-sectional area of Standpipe
Figure 5.5. Das FGE (2005). 2
1loghhe
AkaLt =
Integrate from limits 0 to t
Integrate from limits h1 to h2
after rearranging above equation
after integration
or
Slide 14 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
EMPIRICAL RELATIONSHIPS FOR HYDRAULIC CONDUCTIVITY
210sec)/( cDcmk =
Uniform Sands - Hazen Formula (Hazen, 1930):
Where: c = Constant between 1 to 1.5 D10 = Effective Size (in mm)
eeCk+
=1
3
1
Sands – Kozeny-Carman (Loudon 1952 and Perloff and Baron 1976):
Where: C = Constant (to be determined) e = Void Ratio
85.024.1 kek =
Sands – Casagrande (Unpublished):
Where: e = Void Ratio k0.85 = Hydraulic Conductivity @ e = 0.85
⎟⎟⎠
⎞⎜⎜⎝
⎛
+=
eeCkn
12
Normally Consolidated Clays (Samarasinghe, Huang, and Drnevich, 1982):
Where: C2 = Constant to be determined experimentally n = Constant to be determined experimentally e = Void Ratio
Slide 15 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
EXAMPLE - ESTIMATION OF HYDRAULIC CONDUCTIVITY (NORMALLY CONSOLIDATED CLAYS)
Void Ratio (e) k
(cm/sec) 1.2 0.6 x 10-7
1.52 1.52 x 10-7
⎥⎦
⎤⎢⎣
⎡
+
⎥⎦
⎤⎢⎣
⎡
+=
21
1
22
1
12
2
1
eeC
eeC
kk
n
n
Example 5.6 Das PGE (2005).
GIVEN: Normally consolidated clay with e and k measurements from 1D Consolidation Test.
SOLUTION: Using (Samarasinghe, Huang, and Drnevich, 1982) Equation:
⎟⎠
⎞⎜⎝
⎛⎟⎠
⎞⎜⎝
⎛=2.252.2
52.12.1
sec/52.1sec/6.0 n
cmcmSubstituting
known quantities
REQUIRED: Find k for same clay with a void ratio of 1.4.
Slide 16 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
EXAMPLE - ESTIMATION OF HYDRAULIC CONDUCTIVITY (NORMALLY CONSOLIDATED CLAYS) (continued)
sec/101.14.11
4.1sec)/710581.0(
sec/10581.0
2.112.1sec/106.0
1
75.4
4.1
72
5.47
1
121
cmxcmxk
cmxC
cmx
eeCk
e
n
−=
−
−
=⎟⎟⎠
⎞⎜⎜⎝
⎛
+−=
=∴
⎟⎟⎠
⎞⎜⎜⎝
⎛
+=
⎟⎟⎠
⎞⎜⎜⎝
⎛
+=
Example 5.6 Das PGE (2005).
5.42.252.2
52.12.1
sec/52.1sec/6.0
=∴⎟⎠
⎞⎜⎝
⎛⎟⎠
⎞⎜⎝
⎛= ncmcm n
Slide 17 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
EQUIVALENT HYDRAULIC CONDUCTIVITY IN STRATIFIED SOILS – HORIZONTAL DIRECTION
Figure 5.7. Das FGE (2005).
Considering cross-section of Unit Length 1.
Total flow through cross-section can be written as:
nn HvHvHvqHvq
••++••+••=
••=
1...111
2211
Where: v = Average Discharge Velocity v1 = Discharge Velocity in Layer 1
Slide 18 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
nnnHH
eqeqH
ikvikvikvikv
===
=
;...;; 222111
)(
Figure 5.7. Das FGE (2005).
Substituting v=ki into q equation and using H to
denote Horizontal Direction
Noting that ieq=i1=i2=…=in
)...(1 2211)( nHnHHeqH HkHkHkH
k +++=
Where kH(eq) = Equivalent Hydraulic Conductivity in Horizontal Direction
EQUIVALENT HYDRAULIC CONDUCTIVITY IN STRATIFIED SOILS – HORIZONTAL DIRECTION
Slide 19 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
n
n
hhhh
vvvv
====
====
...
...
21
21
nVnVeqV ikikHhk === ...11)(
Figure 5.8. Das FGE (2005).
Total Head Loss = h h = Sum Head Loss in Each Layer
and
Using Darcy’s Law (v=ki) into v equation and using V to denote
Vertical Direction
Where kV(eq) = Equivalent Hydraulic Conductivity in Vertical Direction
EQUIVALENT HYDRAULIC CONDUCTIVITY IN STRATIFIED SOILS – VERTICAL DIRECTION
Slide 20 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
rhdrdhkq π2⎟⎠
⎞⎜⎝
⎛=
∫∫ ⎟⎟⎠
⎞⎜⎜⎝
⎛=
1
2
1
2
2 h
h
r
rhdh
qk
rdr π
FIELD PERMEABILITY TEST BY PUMPING WELLS UNCONFINED PERMEABLE LAYER UNDERLAIN BY IMPERMEABLE LAYER
Figure 5.9. Das FGE (2005).
)(
log303.2
22
21
2
110
hhrrq
k field−
⎟⎟⎠
⎞⎜⎜⎝
⎛
=πField Measurements Taken:
q, r1, r2, h1, h2
q = Groundwater Flow into Well q also is rate of discharge from
pumping
Equation:
can be re-written as
Solving Equation:
Slide 21 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
rHdrdhkq π2⎟⎠
⎞⎜⎝
⎛=
∫∫ =1
2
1
2
2hh
r
rdh
qkH
rdr π
FIELD PERMEABILITY TEST BY PUMPING WELLS WELL PENETRATING CONFINED AQUIFER
Figure 5.10. Das FGE (2005).
)(727.2
log
21
2
110
hhHrrq
k field −
⎟⎟⎠
⎞⎜⎜⎝
⎛
=Field Measurements Taken: q, r1, r2, h1, h2
q = Groundwater Flow into Well q also is rate of discharge from
pumping Equation:
can be re-written as
Solving Equation:
Slide 22 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
after Casagrande and Fadum (1940) and Terzagi et al. (1996).
SOIL PERMEABILITY & DRAINAGE
Slide 23 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
From FHWA IF-02-034 Evaluation of Soil and Rock Properties.
SOIL PERMEABILITY & DRAINAGE
192
Good drainage
10-8
10-9
10-7
10-6
10-5
10-4
10-3
10-2
10-11.0101
102
10-8
10-9
10-7
10-6
10-5
10-4
10-3
10-2
10-11.0101
102
Poor drainage Practically impervious
Clean gravelClean sands, Clean sand and gravel mixtures
Impervious sections of earth dams and dikes
“Impervious” soils which are modif ied by the effect of vegetation and weathering; f issured, weathered clays; fractured OC clays
Pervious sections of dams and dikes
Very f ine sands, organic and inorganic silts, mixtures of sand, silt, and clay glacial till, stratif ied clay deposits, etc.
“Impervious” soils e.g., homogeneous clays below zone of weathering
Drainage property
Application in earth
dams and dikes
Type of soil
Direct determination
of coefficient of
permeability
Indirect determination
of coefficient of
permeability
*Due to migration of f ines, channels, and air in voids.
Direct testing of soil in its original position (e.g., well points). If properly conducted, reliable; considerable experience required. (Note: Considerable experience
also required in this range.)
Constant Head Permeameter; little experience required.
Constant head test in triaxial cell; reliable w ith experience and no leaks.
Reliable;Little experiencerequired
Falling Head Permeameter;Range of unstable permeability;* much experience necessary to correct interpretation
Fairly reliable; considerable experience necessary (do in triaxial cell)
Computation:From the grain size distribution(e.g., Hazen’s formula). Only applicable to clean, cohesionless sands and gravels
Horizontal Capillarity Test:Very little experience necessary; especially useful for rapid testing of a large number of samples in the f ield w ithout laboratory facilities.
Computations:from consolidation tests; expensive laboratory equipment and considerable experience required.
10-8
10-9
10-7
10-6
10-5
10-4
10-3
10-2
10-11.0101
102
10-8
10-9
10-7
10-6
10-5
10-4
10-3
10-2
10-11.0101
102
COEFFICIENT OF PERMEABILITY
CM/S (LOG SCALE)
Figure 90. Range of hydraulic conductivity values based on soil type.
Another method of estimating hydraulic conductivity empirically through grain size was developed by GeoSyntec (1991) and is presented as Figure 91. As with Hazen’s equation, grain size distribution information is necessary to develop the input parameter (i.e., d15, grain size at 15% passing by weight) of the material. Using this value and the band of values corresponding to gradient and confining stress, a k value can be estimated. This method, as with other empirical methods, can only be used to provide an estimated value within 1 to 2 orders of magnitude of the in-situ condition. The lower the permeability and the higher the variability of grain size in the soil, the higher the error in using empirical relationships based on uniform particle distribution.
Slide 24 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
LAPLACE’S EQUATION OF CONTINUITY Steady-State Flow
around an impervious Sheet
Pile Wall
Consider water flow at Point A:
vx = Discharge Velocity in x Direction
vz = Discharge Velocity in z Direction
Y Direction Out Of Plane Figure 5.11. Das FGE (2005).
x
z
y
Slide 25 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
dxdydzzvv
dzdydxxvv
zz
xx
⎟⎠
⎞⎜⎝
⎛∂
∂+
⎟⎠
⎞⎜⎝
⎛∂
∂+
LAPLACE’S EQUATION OF CONTINUITY Consider water flow at Point A (Soil Block at Pt A shown left)
Rate of water flow into soil block in x direction: vxdzdy
Rate of water flow into soil block in x direction: vzdxdy
Figure 5.11. Das FGE (2005).
Rate of water flow out of soil block in x,z directions:
Slide 26 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
[ ]
0
0
=∂
∂+
∂
∂
=+
−⎥⎦
⎤⎢⎣
⎡⎟⎠
⎞⎜⎝
⎛∂
∂++⎟
⎠
⎞⎜⎝
⎛∂
∂+
zv
xv
ordxdyvdzdyv
dxdydzzvvdzdydx
xvv
zx
zx
zz
xx
LAPLACE’S EQUATION OF CONTINUITY Consider water flow at Point A (Soil Block at Pt A shown left)
Figure 5.11. Das FGE (2005).
Total Inflow = Total Outflow
Slide 27 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
02
2
2
2=
∂
∂+
∂
∂
∴
⎟⎠
⎞⎜⎝
⎛∂
∂−==
⎟⎠
⎞⎜⎝
⎛∂
∂−==
zhk
xhk
zhkikv
xhkikv
zx
zzzz
xxxx
LAPLACE’S EQUATION OF CONTINUITY Consider water flow at Point A (Soil Block at Pt A shown left)
Figure 5.11. Das FGE (2005).
Using Darcy’s Law (v=ki)
Slide 28 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
FLOW NETS – DEFINITION OF TERMS Flow Net: Graphical Construction used to calculate groundwater flow through soil. Comprised of Flow Lines and Equipotential Lines. Flow Line: A line along which a water particle moves through a permeable soil medium. Flow Channel: Strip between any two adjacent Flow Lines. Equipotential Lines: A line along which the potential head at all points is equal. NOTE: Flow Lines and Equipotential Lines must meet at right angles!
Slide 29 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
Figure 5.12a. Das FGE (2005).
FLOW NETS FLOW AROUND SHEET PILE WALL
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14.330 Soil Mechanics Hydraulic Conductivity & Seepage
Figure 5.12b. Das FGE (2005).
FLOW NETS FLOW AROUND SHEET PILE WALL
Slide 31 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
1. The upstream and downstream surfaces of the permeable layer (i.e. lines ab and de in Figure 12b Das FGE (2005)) are equipotential lines.
2. Because ab and de are equipotential lines, all the flow lines
intersect them at right angles. 3. The boundary of the imprevious layer (i.e. line fg in Figure
12b Das FGE (2005)) is a flow line, as is the surface of the impervious sheet pile (i.e. line acd in Figure 12b Das FGE (2005)).
4. The equipontential lines intersect acd and fg (Figure 12b Das
FGE (2005)) at right angles.
FLOW NETS – BOUNDARY CONDITIONS
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14.330 Soil Mechanics Hydraulic Conductivity & Seepage
Figure 5.13. Das FGE (2005).
FLOW NETS Flow under a Impermeable Dam
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14.330 Soil Mechanics Hydraulic Conductivity & Seepage
nqqqq Δ==Δ=Δ=Δ ...321
Figure 5.14. Das FGE (2005).
Δq = k h1 − h2l1
#
$%
&
'(l1 = k
h2 − h3l2
#
$%
&
'(l2 = k
h3 − h4l3
#
$%
&
'(l3 = ...
h1 − h2 = h2 − h3 = h3 − h4 = ... =HNd
Rate of Seepage Through Flow Channel (per unit length):
Using Darcy’s Law (q=vA=kiA)
Potential Drop Where: H = Head Difference Nd = Number of Potential Drops
FLOW NETS Seepage Calculations
Slide 34 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
Figure 5.12b. Das FGE (2005).
d
f
NHN
kq =
If Number of Flow Channels = Nf, then the total flow for all channels per unit length is:
Therefore, flow through one channel is:
dNHkq =Δ
FLOW NETS FLOW AROUND SHEET PILE WALL
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14.330 Soil Mechanics Hydraulic Conductivity & Seepage
GIVEN:
Flow Net in Figure 5.17. Nf = 3 Nd = 6 kx=kz=5x10-3 cm/sec DETERMINE:
a. How high water will rise in piezometers at points a, b, c, and d.
b. Rate of seepage through flow channel II.
c. Total rate of seepage.
Figure 5.17. Das FGE (2005).
FLOW NETS FLOW AROUND SHEET PILE WALL EXAMPLE
Slide 36 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
mmm 56.06
)67.15(=
−
SOLUTION: Potential Drop =
dNH
At Pt a: Water in standpipe = (5m – 1x0.56m) = 4.44m At Pt b: Water in standpipe = (5m – 2x0.56m) = 3.88m At Pts c and d: Water in standpipe = (5m – 5x0.56m) = 2.20m
FLOW NETS FLOW AROUND SHEET PILE WALL EXAMPLE
Figure 5.17. Das FGE (2005).
Slide 37 of 37 Revised 2/2012
14.330 Soil Mechanics Hydraulic Conductivity & Seepage
FLOW NETS FLOW AROUND SHEET PILE WALL EXAMPLE
SOLUTION:
dNHkq =Δ
k = 5x10-3 cm/sec = 5x10-5 m/sec Δq = (5x10-5 m/sec)(0.56m) Δq = 2.8x10-5 m3/sec/m
fd
f qNNHNkq Δ==
q = (2.8x10-5 m3/sec/m) * 3 q = 8.4x10-5 m3/sec/m
Figure 5.17. Das FGE (2005).