Clay Blanket Calculations by iin12614

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```									          Water flow in saturated soil

D A Cameron
Civil Engineering Practice 1

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   1
SEEPAGE – water pressures
Water flows from points of high

[“height of water”] x [w] = water pressure, u

i.e                     h = hT = [he + hp]

Kinetic head is ignored in soils

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Pressure head = height water rises to in a
standpipe above the point

No loss of
Water table level                                        head, h, in soil
hp                       mass,
h             so no flow
he
Arbitrary
datum
Element of soil
within soil mass
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Confined Aquifer
A water bearing layer, overlain and underlain
by far less permeable soils.

Water level in aquifer                                standpipe

Clay, silt
- no free water

x
Clay, silt

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Steady flow in soils – Laminar flow

Assumptions to theory:

• Uniform soil, homogeneous & isotropic
• Continuous soil media
• Small seepage flow (non turbulent flow)
Darcy’s Law of 1850 – a Frenchman

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Darcy’s Law

q = kiA

where               q         =         rate of flow (m3/s)
A         =         area normal to flow
direction (m2)
k         =         coefficient of
permeability (m/s)

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h

Area of
flow, A

Flow rate,
q
Length of flow, l

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Hydraulic Conductivity

• Coefficient of permeability or just “permeability”

• SATURATED soil permeability

Hazen’s formula, for clean, almost uniform sands:

2
d10
k
100
m/sec if particle size in mm

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TYPICAL PERMEABILITIES

Clean gravels                                           > 10-1 m/s

Clean sands, sand-gravel 10-4 to 10-2 m/s

Fine sands, silts                                 10-7 to 10-4 m/s

Intact clays, clay-silts                          10-10 to 10-7 m/s

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Measuring Permeability
[A] Laboratory                                 [A] Laboratory
• Constant head test                             How good is the
• Other

[B] Field                                       [B] Field
• Pumping tests                                 Need to know soil
• Borehole infiltration                         profile (incl. WT) &
boundary conditions
tests

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Lab Test 1: Constant head test

• Cylinder of saturated coarse grained soil
• Water fed under constant head
─ elevated water tank with overflow

• Rate of outflow measured

Repeat the above after raising the water tank

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Water tank -
moveable

ht

q                A
hpB hpC
B
l           he
C
D
soil
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Suitable for clean sands and fine gravels

EXAMPLE:
• If the sample area is 4500 mm2,
• the vertical distance between the 2
standpipe points is 100 mm,
• h is 75 mm
• Outflow is 1 litre every minute

What is the coefficient of permeability?

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Solution
• 1000 cm3/min

OR q = 16.7 cm3/sec = 16.7x10-6 m3/sec

• i = 75/100 = 0.75

• k = q/(iA)

= (16.7x10-6)/(0.75x4500x10-6) m/sec

k = 5 x 10-3 m/sec

Typical permeability of a clean sand or gravel

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   14

For fine sands, silts, & maybe clays
• Rate of water penetration into cylindrical
sample from loss of head in feeder tube
• Must ensure:
− no evaporation
− sufficient water passes through

A slow procedure

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   15
Level at time, t1
Level at time, t2
Tube of cross-sectional
area 'a'

h1
h2

To permeameter cell

Level of cell outflow

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT            16
• Soil sample length, L, & area, A

• Flow in the tube = flow in the soil
– tube has area “a”

 a  L   h1 
k              ln 
A  (t 2 t1 )   h 2 
                

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   17
3. Field testing – drawdown test
Pumping well                                       Water
q
r2                                 table

r1

h2
h1

Impermeable boundary
Drawdown –
phreatic or
flow line
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Drawdown test
Needs
1. a well-defined water table and
2. a confining boundary

Must be able to
1. pull down water table and
2. create flow
(phreatic line = uppermost flow line)

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   19
Solution

Axi-symmetric problem

By integration of Darcy’s Law,

     q          r2 
k              ln  
 π(h 2 h 2 )   r 
     2   1   1 

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   20
TUTORIAL PROBLEMS
A canal and a river run parallel, an average of 60 m
apart. The elevation of water in the canal is 200 m
and the river 193 m. A stratum of sand intersects both
the river and canal below the water levels

The sand is 1.5 m thick and is sandwiched between
strata of impervious clay

Compute the seepage loss from the canal in
m3/s per km length of the canal, given the
permeability of the sand is 0.65 mm/s

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   21
THE PROBLEM

Sand
seam

RL 200 m                          RL 193 m

canal
river
60 m

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SOLUTION

q = kiA
k = 0.65 mm/s = 0.65 x 10-3 m/s
h = 7 m

q = 0.65 x 10-3 x 0.117 x 1.5 m2/m length
q = 0.114 x 10-3         m3/sec /m length
q = 0.114               m3/sec/km length

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Hydraulic

RL 200 m                          RL 193 m

h = 7 m

l = 60 m

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Flow Lines
– shortest paths for water to exit

Equipotential lines

hp1                                          h

Flow
tube                                                       hp2

h1e1                  l                     he2

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The Flow Net
- FLOW LINES

Run  parallel to impervious boundaries
(impermeable walls or “cut-offs”) and the
phreatic surface
The “Phreatic surface” is the top flow line

2 consecutive flow lines constitute
a “flow tube”

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   26
h

5 Flow Lines

Impervious boundary

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The Flow Net
- EQUIPOTENTIALS
• Are lines of equal total head

• The total head loss between consecutive
equipotentials is constant

• Equipotentials can be derived from
boundary conditions and flow lines

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   28
Flownet Basics
Water flow follows
paths of maximum
h i
imax                                             i m ax 
b m in
 flow lines and
equipotentials must
cross at 90°, since:

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT       29
Since q is the same, ratio of
sides will be constant for all
the “squares” along the flow
tube

h

M

Equi- potential lines
Impervious boundary
DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   30
 h i 
Flow
q
q  k      a  per m
 b 
q  constant,if a : b  is constant

hi  head lost between equipotent
ials

b
a
Common convention
draw “squares” with a = b

“square, M”, a x b

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT          31
Discharge in flow direction,
Equipotentials                            = q per “flow tube”

h3

90º

l
h2
Flow lines

h1

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT        32
Flownet Construction

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Flow Net Calculations
Nd equal potential drops along length of flow?
Then the head loss from one line to another is:
h1-2 = (h) = h / Nd

From Darcy’s Law, flow rate in a flow tube,
 Δh i 
ΔqkiA k      a1
 b 
OR
 Δh  a 
 N  b 
Δqk     
 d
DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   34
Flow Net Calculations

BUT              a=b
AND              total flow for Nf “flow channels”,
per unit width is:

Nf
qk    Δh
Nd

But only for “squares”!

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   35
Example: if k = 10-7 m/sec, what would be the
flow per day over a 100 m length of wall?

Dam
cutoff
50 m of water
5 m of water

Low permeability rock

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Calculations

Nd = 14
h = 45 m                          = 10-7(5/14)45 x 100 m length
k = 10-7 m/sec                     = 0.000161 m3/sec
= 13.9 m3/day

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT     37
Example: what is the hydraulic gradient
in the “square” C?

Dam
cutoff
50 m of water
5 m of water

Low permeability rock

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT           38
Calculations

[h / Nd] = 45/14                         Answer: 1.1 and
= 3.2 m                           therefore

Average length of

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT     39
The value of i for which the effective
stress in the saturated system becomes
ZERO!

Consequences:

no stress to hold granular soils together

 soil may flow 

“boiling” or “piping” = EROSION

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   40
Seepage Condition – upward flow of water
 = satz = total stress
u due to seepage,
B
= i(z)(w)
z
(represents proportion of h
A                                  occurring over length AB)
 =  - u
= (satz) – (wz + i(z)w)
 = z – i(z)w
 = 0, when           z = i(z)w         OR        i = (/ w)

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT            41
Likelihood of Erosion
GRANULAR SOILS chiefly!
When the effective stress becomes zero,
no stress is carried by the soil grains

Note: when flow is downwards, the effective
stress is increased!

So the erosion problem and ensuing
instability is most likely for upward flow,
i.e. water exit points through the foundations of
dams and cut-off walls

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   42
DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   43
Minimising the risk of erosion
1. Add more weight at exit points

permeable concrete mats?

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT         44
Lengthen flow path?
1. Deeper cut-offs
2. Horizontal barriers
3. Impermeable blanket on exit surface

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   45
Simple cut-offs (FESEEP)

Nf = 5
Nf
Nd =10= 5
NN=115
d f=

Nd =13

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT    46
“Impermeable” Clay Blanket

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   47
Summary: Key Points
•   Darcy’s Law
•   Coefficient of permeability
•   Measurement of permeability
•   Flownets
•   Flownet rules
•   Seepage from flownets
•   Piping, boiling or erosion

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   48
Exercises
a) Draw a flow net for seepage under a vertical sheet pile
wall penetrating 10 m into a uniform stratum of sand
20 m thick.

b) If the water level on one side of the wall is 11 m above
the sand and on the other side 1.5 m above the sand,
compute the quantity of seepage per unit width of wall.
[k = 3  10-5 m/s]

c) What is the factor of safety against developing the
“quick” condition on the outflow side of the wall?
[sat= 21 kN/m3]

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   49
Finite Difference
and other numerical approaches

Authors:
Mahes Rajakaruna (ex UniSA)
& University of Sydney (FESEEP)

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   50
Finite Difference approach to flow nets
- flow line set up
ROWCO   A     B   C   D   E    F   G   H   I   J   K   L   M   N   O   P   Q   R   S     T   U   V    W
L

1    100   Soil level                                                           104

2    100                                                                        104

3    100                                                                        104

4    100
Cell H5                                         104

5    100                                                                        104

6    100                                                                        104

7    100                                                                        104

8    100                                                                        104

9    100                   Interior cell value =                    104 104 104 104 104 104 104

10    100                   (H4+I5+H6+G5)/4
11    100

12    100
Impermeable boundary
13    100

14    100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT                                51
Flow lines from finite difference

98-99
99-100
100-101
101-102
102-103

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT      52
Equipotentials from finite difference

113-114
112-113
111-112
110-111
109-110
108-109
107-108
106-107
105-106
104-105
103-104
102-103
101-102
100-101

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT    53
FESEEP: University of Sydney
cutoff

Mesh of foundation soil

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   54
FESEEP Output
(University of Sydney)
flownet

pore pressures                       increasing

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   55
Dam
cutoff
50 m of water
5 m of water

Low permeability rock

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT             56

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