Water flow in saturated soil

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Water flow in saturated soil

• D A Cameron
• Introduction to Civil and Mining Engineering

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SEEPAGE water pressures

• Water flows from points of high to low
• TOTAL head
• WATER HEADS
• head of water x ?w water pressure
• Total head elevation head pressure head
• i.e h (or hT) he hp
• Kinetic head is ignored in soils

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Head of Water

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

No loss of head, h, in this soil mass, so no
flow - Steady State
Water table level
hp
h
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.

standpipe
Water level in aquifer
Clay, silt - no free water
Sand aquifer
Clay, silt
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Steady flow in soils Laminar flow

• Assumptions to theory
• Uniform soil, homogeneous and isotropic
• Continuous soil media
• Small seepage flow (non turbulent flow)
• Darcys Law of 1850

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Darcys Law

• q kiA
•  
• where q rate of flow (m3/s)
• i hydraulic gradient
• A area normal to flow direction (m2)
• k coefficient of permeability (m/s)

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Hydraulic Gradient, i
?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!
• Hazens formula, for clean and almost uniform
  sands

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TYPICAL PERMEABILITIES

• Clean gravels gt 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
• Constant head test
• Falling head test
• Other
• B Field
• Pumping tests
• Borehole infiltration tests

A Laboratory How good is the sample? B
Field Need to know soil profile (incl. WT) and
boundary conditions
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1. Constant head permeameter
Water tank - moveable
overflow
?ht
A
hpC
hpB
q
B
?he
C
D
soil
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Constant head test

• 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
• Typically a clean sand or gravel permeability

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2. Falling head permeameter
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Falling head test

• Suited to low permeability materials
• silts and clays
• Soil sample length, L, and area, A
• Flow in the tube flow in the soil

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3. Field testing drawdown test
Pumping well
q
r2
Water table
r1
h2
h1
Impermeable boundary
Drawdown -phreatic line
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Drawdown test

• Needs
• a well-defined water table
• and confining boundary
• Must be able to
• pull down water table
• and create flow
• (phreatic line uppermost flow line)

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Solution

• Axi-symmetric problem
• By integration of Darcys Law,

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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.

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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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THE PROBLEM
Hydraulic gradient, i 0.117
RL 200 m
RL 193 m
?h 7 m
l 60 m
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Flow Lines shortest paths for water to exit
Phreatic surface
Equipotential line-
?h
hp1
Stream tube
hp2
he1
Dl
he2
1
Head reference line
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Equipotentials

• Are lines of equal total head
• Can be derived from boundary conditions
• and flow lines

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The Flow Net
Flow lines - run parallel to impervious
boundaries and the phreatic surface. Phreatic
surface the top flow line Equipotential lines
- line of constant total head - the total head
loss between consecutive equipotentials is
constant 2 consecutive flow lines constitute
a flow tube
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Flownet Basics

• Water flow follows paths of maximum hydraulic
  gradient, imax
• flow lines and equipotentials must cross at 90,
  since
• imax ?(?h) / bmin

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Since ?q is the same, a/b will be constant for
all the squares along the flow tube
Flow ?q
?h
Flow Lines
?(?h)
M
b
a
square M
Equi- potential lines
Impervious boundary
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Discharge in flow direction, ?q / flow tube
Equipotentials
h3
90º
l
h2
Flow lines
h1
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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
• ?(?h) ?h / Nd
• From Darcys Law

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Flow Net Calculations

• BUT a b
• AND total flow for Nf flow channels,
• per unit width is  

But only for squares!
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Example if k 10-7 m/sec, what would be the
flow per day over a 50 m length of wall?
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Calculations

• Answer 6.72 m3

• Nf 3 or 4
• Nd 9 or 10?
• ?h 35 m?
• k 10-7 m/sec

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Example what is the hydraulic gradient in the
square C?
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Calculations
Answer 0.14

• ?h / Nd 35/9
• 3.9 m head / drop
• Average length of flow is about 23 m

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Finite Difference spreadsheet solution

• Author Mahes Rajakaruna
• emailed to students today

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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
Soil level
1
100
104
2
100
104
3
100
104
Cell H5
4
100
104
5
100
104
6
100
104
7
100
104
8
100
104
Interior cell value (H4I5H6G5)/4
9
100
104
104
104
104
104
104
104
10
100
11
100
Impermeable boundary
12
100
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
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Flow lines from finite difference program
(spreadsheet)
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Equipotentials from finite difference program
(spreadsheet)
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Other numerical approaches FESEEP
cutoff
Mesh of foundation soil
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FESEEP Output (University of Sydney)
flownet
increasing
pore pressures
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Critical hydraulic gradient

• 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!

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Seepage Condition upward flow of water

• ?satz total stress
• ?u due to seepage
• iABz(?w)
• (represents ?h ?hp)
• ?? ? - u
• (?satz - ?wz) - iz?w
• ?? ??z - iz?w

B
z
A
?? 0, when ??z - iz?w 0 OR i
??/ ?w
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Critical Hydraulic Gradient
Free surface, end area , A
h2
L
Small cylindrical element of soil
h1
FW
Flow direction
Seepage force, Fs
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Critical hydraulic gradient

• Fs ?h?wA
• (h1 h2) ?wA
• Fw (?sat - ?w)AL
• (??)AL
• Equating the 2 forces
• i ??/ ?w as before

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Likelihood of Erosion
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
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Minimising the risk of erosion

• 1. Add more weight at exit points

permeable concrete mats?
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Lengthen flow path?
1. Deeper cut-offs 2. Horizontal barriers 3.
Impermeable blanket on exit surface
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Summary

• Heads in soil
• Darcys Law
• Coefficient of permeability
• Measurement of permeability
• Flownets
• Flownet rules
• Seepage from flownets
• Piping, boiling or erosion
• Critical hydraulic gradient

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• 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
• What is the factor of safety against developing
  the quick condition on the outflow side of the
  wall? ?sat 21 kN/m3

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