CE222 SM 10 Seepage and Flow Nets

Soil Mechanics ICE‐222CE 222

Water in soil: Seepage and flow nets

1

Teton dam failure sequence 2

Newly completed Teton Dam as it appeared in mid May 1976, as the reservoir wasfilling at the rate of 3 feet per day. The rate of filling is usually limited to no morethan 1 foot per day. This view is looking towards right abutment.

Mid May 1976Mid May 1976Newly completed Teton DamNewly completed Teton Dam


Leakage was initially noted around 7:00 AM on Saturday June 5, 1976. This view shows a dozer being sent down to fill in the hole at elevation 5200 around 10:45 AM

June 5, 1976June 5, 1976,,10:45 am10:45 am


The dozer is lost in the expanding hole, around 11:20 AM on June 5th. Note turbid nature of outflow along the abutment.

June 5, 1976June 5, 1976,,11:20 am11:20 am


Rapidly deteriorating situation as it appeared around 11:30 AM. A massive hole has developed in the downstream face of the embankment and is migrating upward.

J 5 1976J 5 1976June 5, 1976June 5, 197611:30 am11:30 am


The hole continues to enlarge and rise toward the crest of the right abutment. This is about 11:50 AM.

June 5, 1976June 5, 197611:50 am11:50 am


Dam crest beginning to breach at 11:55 AM on Saturday June 5, 1976. Note increasing discharge.

June 5, 1976June 5, 197611:55 am11:55 am


Maximum flood discharge emanating from gap in dam’s right abutment, just after noon on June 5th , 1976.

June 5, 1976June 5, 1976,,After noonAfter noon

Present day Teton dam site 9

Seepage Terminologyp g gyStream line is simply the path of a water molecule.From upstream to downstream, total head steadily decreases p , yalong the stream line.

datum

hL

concrete dam TH = 0TH = hL

impervious strata

soil

Equipotential line is simply a contour of constant total head.

concrete damdatum

hL

concrete damTH = 0TH = hL

TH=0 8 h

impervious strata

soilTH=0.8 hL

FlownetFlownetA network of selected stream lines and equipotential lineslines.

concrete damconcrete dam

curvilinearcurvilinear square

90º

impervious strata

soil90

Quantity of Seepage (Q)

fL N

NkhQ = ….per unit length normal to the plane

# of flow channels

dN# of equipotential drops

hL

head loss from upstream to downstream

concrete dam

impervious strata

Heads at a Point XHeads at a Point XTotal head = hL - # of drops from upstream x Δh

Elevation head = -z LhPressure head = Total head – Elevation head d

L

N=

datumhL

concrete dam

datum

z

TH = hL TH = 0

X

z

Δh

impervious strata

Flow net 15

A Flow Net consists of two groups of curves:

Equipotential lines: Equipotential lines are lines that pass through points f l h dof equal head.

Flow lines: Flow lines (aka stream lines) represent the path that a particle of water takes as it travels through the soil massof water takes as it travels through the soil mass.

02

2

2

2

=∂∂

+∂∂

zh

xh Laplace’s Laplace’s

Equation of Equation of

Equipotential lines

∂∂ zx ContinuityContinuity

Equipotential lines

Flow lines

Flow net – Equipotential lines 16

The space between two adjacent equipotential lines represents a drop in head.

The space between two adjacent equipotential lines is called an equipotential space.

Flow net – Flow lines 17

Th b t t dj t fl li i ll d flThe space between two adjacent flow lines is called a flow path

Flow lines

Flow net 18

An equipotential line means potential head at all points is equal (i.e. total head is constant).

Water in a piezometer (placed at different points along an equipotential line) will rise to the same elevation.

Δh

h

Flow net 19

An equipotential line

Impounded water

means potential head at all points is equal (i.e. total head is

Tail water

constant).

Water in a piezometer p(placed at different points along an equipotential line) will q p )rise to the same elevation.

Flow net 20

A combination of flow lines & equipotential lines is called flow net

Impounded waterTail water

Impounded water

Flow net under a dam 21

ImpoundedImpounded water Tail water

Flow net under a dam with toe filter 22

Flow net under a dam with sheet pile 23

Seepage and flow net through dam 24

25

Head: 10 m

10 m 10 m20 m

10 mKx = 1.0 e -005 m/seck k

Seepage analysis using SEEP/W

kx = ky

Head: 1 m 21 30

22 29

20 m

23

24 27

28

3.33

77e-

005

25 26

40 m

2610 m 20 m 10 m

Head: 10 m

10 mKx = 1.0 e -005 m/secKx = Ky

Seepage analysis using SEEP/W

Head: 1 m

Kx = Ky

21 30

22

29

20 m

23

7

28

2.4

766e

-005

24 25 26

27

40 m

Sample flow nets 27

Sample flow nets 28

Sample flow nets 29

Construction of flow nets 30

• Equipotential lines intersect flow lines at right angles (there is no flow along an equipotential line, therefore, all of the flow must be at 90° to it)must be at 90 to it)

• Just like contour lines, flow lines cannot cross other flow lines & equipotential lines cannot cross other equipotential linesequipotential lines cannot cross other equipotential lines.

• The flow elements form an approximate “curvilinear square”approximate curvilinear square . Although the sides may curve, a curvilinear square is as broad as it is long, so that a circle inscribed in it touches all four sides.

H h i i l• Hence, each equipotential space must represent an equal drop in head.


The upstream and downstream surfaces of the permeableof the permeable layer (lines ab and de) are equipotential llines.

Since ab & de are equipotential lines, all the flow lines intersect them atintersect them at right angles.


The boundary of the impervious layer (line fg and line acd) arefg and line acd) are flow lines,

f dSince fg & acd are flow lines, all the equipotential lines q pintersect them at right angles.

Seepage calculation from flow net 33

In any flow net, the strip between any two adjacent flow lines is called flow channel.

Since there is no flow across the flow linesSince there is no flow across the flow lines, Δq1 = Δq2 = Δq3 = Δq


⎞⎛⎞⎛From Darcy’s law ( ) ( ) L=×⎟⎟

⎠

⎞⎜⎜⎝

⎛ −=×⎟⎟

⎠

⎞⎜⎜⎝

⎛ −==Δ 11 2

2

321

1

21 ll

hhkll

hhkkiAq

If flow elements are approx. squares, the drop in piezometric level between any two adjacent equipotential lines is same This isadjacent equipotential lines is same. This is called equipotential drop.


If flow elements are approx. squares, the drop in piezometric level between any two adjacent equipotential lines is same. This is called equipotential drop.

dNHhhhhhh ==−=−=− L433221

called equipotential drop.

dNHkq =Δ where H = head

difference between upstream andupstream and downstream sides

Nd = number of potential drops


If number of flow channels in a flow net is equal to Nf, the total flow rate through all the channels per unit length can be given by

fd

NNHkq =

Example 37

A flow net for flow around a single row of sheet piles in a permeable soil layer is shown. Given that kx = kz = k = 4.2 × 10–6 cm/sec, determine

1) How high (above the ground surface) the water will rise if piezometers are placed at points a, b, c, and d.

2) h f2) The rate of seepage through flow channel II per unit length (perpendicular to the section shown.

Flow nets in anisotropic soil 38

02

2

2

2

=∂∂

+∂∂

zhk

xhk zx

For anisotropic soils, kx ≠ kz. In this case, the equation represents two families of curves that do not meet at 90o. However, we can rewrite

22 ∂∂ hh( ) 02

2

2

2

=∂∂

+∂

∂zh

xkkh

xz

Substituting ( )xkkx xz=′

02

2

2

2

=′∂

∂+

′∂∂

zh

xh

Flow nets in anisotropic soil 39

02

2

2

2

=′∂

∂+

′∂∂

zh

xh

fd

zx NNHkkq =

To construct the flow net, use the following procedure:

1 Adopt a vert scale for drawing the cross section1. Adopt a vert. scale for drawing the cross section.

2. Adopt a horiz. scale such that horiz. scale vert. scale.×= xz kk

3. With scales adopted in steps 1 & 2, plot the vertical section through permeable layer parallel to the direction of flow.

4. Draw the flow net for permeable layer on the section obtained from step 3, with flow lines intersecting equipotential lines at right angles and elements as approx. squares.

40

Transformed Transformed sectionsection

True sectionTrue section

Uplift pressure 41

120

Uplift pressure under dam

100

120

Without cutoff

With cutoff

60

80

Gradien

t

40

Y‐

0

20

0 5 10 15 20 25

Distance (m)

Seepage through an earth dam on an impervious base

42

Dupuit 1863

Casagrande (1932)Casagrande (1932)

Seepage through an earth dam on an impervious base

43

A step‐by‐step procedure to obtain the seepage rate q (per unit length of the dam) is as follows:

1. Obtain α.

2. Calculate Δ (see (figure below) and then 0.3 Δ.

3. Calculate d.

4. With known values of α and d, calculate L.

5. With known values of L, calculate q. α2sinkLq =

Staged dewatering through well‐points 44

Illustrations of flow nets

Trench supported by sheet pilesTrench supported by sheet piles5m

6m

6m

Uniform sand6m

Impermeable clay

6m

Impermeable clay


6m

6m

Uniform sand6m

Impermeable clay

6m

Impermeable clay


6m Δh=6m

6m

Nh=10Nf=2.5+2.5

6m Uniform sand

Impermeable clay

6m

Impermeable clay

Excavation supported by a sheet pileExcavation supported by a sheet pileSteel sheet

Water pumped away

Uniform sand

Shale

Excavation supported by a sheet pileExcavation supported by a sheet pileSteel sheet

Water pumped away

Uniform sand

Shale

Reduced sheet penetration; possible liquefaction ′ = 0liquefaction σ′v = 0

Steel sheet

Uniform sand

Shale

Reduced sheet penetration; possible liquefaction ′ = 0liquefaction σ′v = 0

Reservoir Tail water

U if dUniform sand

Shale

Concrete dam or weir

Reservoir Tail water

U if dUniform sand

Shale

Concrete dam with cut-off; reduces uplift pressure

Reservoir

U if dUniform sand

Shale

Concrete dam with cut-off; reduces uplift pressure

Reservoir

U if dUniform sand

Shale

Pumped well in confined aquiferObservation wellpumped wellElevation

Aquifer heads

H

D aquiferRadial flow

Impermeable stratum

Plan

Pumped well in confined aquiferObservation wellpumped wellElevation

Aquifer heads

H

D aquiferRadial flow

Impermeable stratum

Plan

Clay dam, no air entry

reservoiratmospheric line

reservoir

drain

clay

Shale

Clay dam, no air entry

atmospheric line

drainreservoir

clay

Shale

Clay dam, no air entryObservation well

atmospheric line

drainreservoir

clay

Shale

Clay dam, no air entry, reduced drain; seepage out of downstream faceseepage out of downstream face

atmospheric lineNot possibleNot possible

reservoir

clay

Shale

Clay dam, with air entry

reservoirreservoir

drain

clay

Shale

Clay dam, with air entry

reservoirreservoir

drain

clay

Shale

Clay dam, no capillary, reduced drain; seepage out of downstream faceseepage out of downstream face

reservoirreservoir

clay

Shale

Clay dam, no capillary, reduced drain; seepage out of downstream faceseepage out of downstream face

reservoirreservoir

clay

Shale

Documents

CE222 SM 10 Seepage and Flow Nets