4. Soil Permeability and Seepage-part 2 - UniMasr · In this case, factor of safety against...

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4. Soil Permeability and Seepage

Soil Mechanics2010 - 2011

Dr. Manal A. Salem – Soil Mechanics

Critical Hydraulic Gradient (icr)Water level is not the same on both sides, upward flow

At point X:

Total downward force = γwhw + γsatL

Total upward force = γw(L+hw+h)

If total downward force = total upward force critical condition, at which:γwhw + γsatL = γw(L+hw+h)γsatL = γwL+γwhγsatL – γwL = γwhγsubL = γwh

L

x

Soil

hw

h

Critical hydraulic gradient = icr = h/L = γsub/γw

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Critical Hydraulic Gradient (icr)If i > icr, unstable condition, the soil particles tend to move:

Boiling, quick condition SandHeave Clay

In this case, factor of safety against boiling, quick condition, or heave <1.0

where:

F.S against boiling = icr/i

=

L

x

Soil

hw

h

hL

w

sub

γγ


Two Dimensional FlowModeled by Laplace’s equation of continuityQx = Avx = dy x 1 x vx

Qy = Avy = dx x 1 x vy

Assuming no volume change:dQx + dQy = 0

Qy

Qy+dQy

dx

dz=1

dy

xy

z

Qx Qx+dQx

xvdydxdx

xvAdQ xx

x ∂∂

=∂∂

=

yv

dxdydyyv

AdQ yyy ∂

∂=

∂

∂=

0=∂

∂+

∂∂

yv

dxdyxvdydx yx

Equation 10=∂∂

+∂∂

yv

xv yx

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Two Dimensional Flow

Substitute in Equation 1:

For isotropic conditions: kx = ky = k

Qy

Qy+dQy

dx

dz=1

dy

xy

z

Qx Qx+dQx

xhkikv xxxx ∂∂

==

2

2

xhk

xv

xx

∂∂

=∂∂

yhkikv yyyy ∂∂

==

2

2

yhk

yv

yy

∂∂

=∂

∂

0=∂

∂+

∂∂

yv

xv yx

02

2

2

2

=∂∂

+∂∂

yhk

xhk yx

02

2

2

2

=∂∂

+∂∂

yh

xh

Laplace’s Equation of Continuity


Two Dimensional Flow

Solution:

1. Analytical

2. Graphical flow net

02

2

2

2

=∂∂

+∂∂

yh

xh

Laplace’s Equation of Continuity

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Graphical solution of Laplace’s equation of continuity

1st Application: Flow net around cut-off wall or sheet pile:

+∞-∞

Sheet pileHWL

LWLGS

Impermeable boundary

Flow Nets

= Datum

H

b 0H ed

c

a

f g

H-Δh

H-2Δh

Δh

2Δh


Boundary Conditions:bcd and fg flow lines (1st and last )ab and de equipotential lines (1st and last )

Flow net:Two orthogonal families of curves:1. Flow lines: along which a water particle travels from

upstream to downstream2. Equipotential lines: lines of constant total head

piezometers reach same level

Conditions:1. Lines intersect at right angles2. Square elements3. Flow lines don’t intersect4. Equipotential lines don’t intersect

Need to draw to scale

Flow Nets

Impermeable

e

c

g

a db

f

5


Rate of seepage or flow “Qt”:nf = number of flow channels(ex: 5)nd = number of head drops(ex: 11)For a single flow channel:

Qi = Av= Aki= b.1.k(Δh/b)

Δh = h1-h2 = H/nd : H = total head difference between upstream and downstreamQi = kH/nd

For nf flow channels:Qt = nfQi

= (nf/nd)kH

Flow Nets

bb

h1

h2

Qi

unity

H


Factor of safety against piping:Upstream: water movement ( ) in same direction of soil weight ( )Downstream: water movement ( ), soil weight ( ) check icr?Most critical hydraulic gradient is checked at “exit square”iexit = Δh/Lexit : Δh=H/nd

F.S. against piping = icr/iexit >1.2where, icr = γsub/γwIn the shown example, if F.S. against piping<1.2, increase penetration depth of sheet pile

Flow NetsH

H Lexit

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Flow Nets 2nd Application: Flow net under Dam/Weir:

Boundary Conditions:bcde and gh flow lines (1st and last )ab and ef equipotential lines (1st and last )

Impermeable boundary+∞-∞

g h

HWL

LWL

H

= Datum

a b

c d

ef

0

H

H-Δh

H-2Δh

Δh

2Δh

GS

GS


Flow Nets Rate of seepage (flow) “Qt”:

nf = 3 (in this example)nd = 13 (in this example)Qt = (nf/nd)kH

F.S. against piping = icr/iexit>1.2, where:

iexit = Δh/Lexit : Δh=H/nd

icr = γsub/γw

H

Lexit

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Flow Nets

F.S. against uplift or floating:

F.S. = weight of weir ÷ uplift force (U) >1.2

Uplift force (U) = area of uplift pressure (u) diagramcalculate (u) at 2 points (c and d)Uplift pressure (u) = (Total head - position head) x γw

If unsafe: Increase length of weir or put sheet pile at weir base?DatumFlow net


g h

HWL

LWL

H

= Datum

a b

c d

ef

0

H

H-Δh

H-2Δh

Δh

2Δh

GS

GS


Flow Nets 3rd Application: Flow net under weir with sheet pile:

Boundary Conditions:bcdefgi and kl flow lines (1st and last ) ab and ij equipotential lines (1st and last )

Impermeable boundary+∞-∞ k l

HWL

LWL

H

= Datum

a b

c g

ij

0

H

H-Δh

H-2Δh

Δh

2Δh

d

e

f

GS

GS

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Rate of seepage (flow) “Qt”:

nf = 3 (in this example)nd = 15 (in this example)Qt = (nf/nd)kH

F.S. against piping = icr/iexit>1.2, where:

iexit = Δh/Lexit : Δh=H/nd

icr = γsub/γw

Flow Nets

Hi

Lexit


Flow Nets

F.S. against uplift or floating:

F.S. = weight of weir ÷ uplift force (U) >1.2

Uplift force (U) = area of uplift pressure (u) diagramcalculate uplift pressure at 4 points along base (a, d, f, g)Area of uplift pressure diagram F.S. against uplift


k l

HWL

LWL

H

= Datum

a b

c g

ij

0

H

H-Δh

H-2Δh

Δh

2Δh

d

e

f

4. Soil Permeability and Seepage-part 2 - UniMasr · In this case, factor of safety against...

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