Table of Content - LSU Hurricane Engineeringhurricaneengineering.lsu.edu/CourseMat/CE_4780_-_Seepage.pdf5 Flow Nets The procedure consists on drawing a set of perpendicular lines:

1

CE 4780 Hurricane Engineering II

Section onFlooding Protection:

Earth Retaining Structures and Slope Stability

Dante FrattaFall 2002

Table of Content

IntroductionShear Strength of SoilsSeepage Analysis Methods of Slope Stability Analysis Design of Earth Retaining Structures Example Problems

Three weeks of classes

2

Seepage Analysis

Introduction

Flow is governed by the total head!!

Swimming pool

Datum

hz1

hp1

hT1

hp2

hz2

hT2

Datumhz1

hp1

hp2

hz2

hT2hT1

Open Channel Filter

Datumhp2

hp2

Seepage Analysis

ObjectivesTo obtain pore pressure (stability analysis)To calculate flowTo verify piping conditions

Retaining wall Cofferdam

Sheet pile

Drainage pipe

Hydraulic dam

3

Laplace’s Equation

Elemental Cube:Saturation S=100 %Void ratio e= constantLaminar flow

Continuity:

0dyqdxqqq yq

yxq

xyxyx =⎟

⎠⎞⎜

⎝⎛ +++−+ ∂

∂∂∂

0dydx yq

xq yx =+ ∂

∂∂∂

outin qq =

xq dxq xq

xx

∂∂+

dyq yq

yy

∂∂

+

yq

dx

dy


Continuity:

Darcy’s law:

Replacing:

if kx=ky Laplace’s Equation!(isotropy):

0dydx yq

xq yx =+ ∂

∂∂∂

1dykAikq xh

xxxT ⋅⋅⋅=⋅⋅= ∂

∂

1dxdyk1dydxk0 2T

2

2T

2

yh

yxh

x ⋅⋅⋅+⋅⋅⋅=∂

∂

∂

∂

2T

2

2T

2

yh

yxh

x kk0∂

∂

∂

∂ ⋅+⋅=

2T

2

2T

2

yh

xh0

∂

∂

∂

∂ +=

4


Typical cases1 Dimensional:

linear variation!!

2-Dimensional:

3-Dimensional:

2T

2

xh0

∂

∂=

ittancons xhT == ∂∂

xbahT ⋅+=

2T

2

2T

2

yh

xh0

∂

∂

∂

∂ +=

2T

2

2T

2

2T

2

zh

yh

xh0

∂

∂

∂

∂

∂

∂ ++=

Laplace’s Equation Solutions

Exact solutions (for simple B.C.’s) Physical models (scaling problems)Graphical solutions: flow netsAnalogies: heat flow and electrical flowNumerical solutions: finite differencesApproximate solutions: method of fragments

5

Flow NetsThe procedure consists on drawing a set of perpendicular lines: equi-potentials and flow lines.These set of lines are the solution to the Laplace’s equation.It is an iterative (and tedious!) process.Identify boundaries:

First and last equi-potentialsFirst and last flow lines

Flow Nets

gradient:

flow per channel:

total flow:

Flow channel

Equipotential lines b= ∆l

a

∆q

∆h= equipotential drop

bbh

lhi eN

h=∆=

∆∆=

Ab

kAlhkq eN

h⋅⋅=⋅

∆∆⋅=∆

e

ff N

NbahkNqq ⋅⋅⋅=⋅∆=

6

Flow Nets– Confined FlowExample

Scale: 1 square = 4 m2

A B

11.0 m

10.0 m

2.0 m16.0 m

permeabilityk = 10-6 m/s

h= 9.0 m

exit gradient

Datum

Flow Nets – Confined FlowExample

11.0 m

10.0 m

2.0 m16.0 m



Last flow line

First flow lineFirst equipotential Last equipotential

7


11.0 m

10.0 m

2.0 m16.0 m



11.0 m


A B10.0 m

16.0 m



2.0 mexit gradient

8


Seepage loss under the dam:

Exit gradient:

msm107.2

103m9

sm10

NNhkq

366

e

f⋅

⋅=⋅⋅=⎟⎟⎠

⎞⎜⎜⎝

⎛⋅∆⋅= −−

45.0m2m9.0

llhi eN

h

e ==∆

=∆∆

=

( ) !!pipinglh1i wsatwcrit ⇒γ−γ⋅∆=γ⋅∆∴=

Flow NetsPiping

(from McCarthy 1998)

9

Flow NetsExample

Total head at points A and B:

Pressure head at points A and B:

kPa121m1.12m8m1.20hhh zATAPA ⇒≈=−=−=

m1.20101m9m21

NNhhh

f

AToTA =⋅−=⋅−=

m5.16105m9m21

NNhhh

f

BToTB =⋅−=⋅−=

kPa85m5.8m8m5.16hhh zBTBPB ⇒≈=−=−=

Flow Nets – Unconfined Flow

Example:

10

Flow Nets - Unconfined Flow

Line of Seepage

(from Harr 1990)

Electrical Flow Analogy

Electric conductionOhm’s law:

Resistance:

Darcy’s law:

RVIRIV =∴⋅=

ALR ⋅ρ=

ALV1I ⋅⋅

ρ=

Aikq h ⋅⋅=

Ai1I e ⋅⋅ρ

=

11

Electrical Flow Analogy

(from Coduto 1999)

Finite Differences Method

Laplace’s equation:

Finite differences:(Taylor’s series expansions)

2T

2

2T

2

yh

xh0

∂

∂

∂

∂ +=

( )1ii

TTT

Tx

h hhx

1x

h−

−⋅∆

=∆∆≈∂

∂

x

hT(x)

∆x

∆hT

xi-1 xi

hTi-1

hTi

( )1ii1i2

T2

TTT2xh hh2h

x1

−++⋅−⋅

∆≈

∂

∂

12


Laplace’s equation in finite differences:

Solving for hTi,k (if ∆x = ∆y):

or

( )1k,i1k,ik,1ik,1ik,i TTTT4

1T hhhhh

−+−++++⋅=

( )topbottomrightleftcenter TTTT4

1T hhhhh +++⋅=

( ) ( )1k,ik,i1k,i2k,1ik,ik,1i2 TTTy

1TTTx

1 hh2hhh2h0−+−+

+⋅−⋅++⋅−⋅=∆∆

hTcenter

hTbottomr

hTtop

hTleft hTright


11.0 m

8.0 m

2.0 m



Datum

1 2 3

5

8

6

4

10

7

9

Number Head Head Top BottomLeft Right Head1 19 8 1 1 1 1 192 19 8 2 2 2 2 193 10 8 3 3 3 3 104 10 8 4 4 4 4 105 0 4 1 8 6 6 4.756 0 4 2 9 5 7 4.757 0 4 4 10 6 6 2.58 0 0 5 5 9 9 09 0 0 6 6 8 10 0

10 0 0 7 7 9 9 0

13

Finite Differences MethodExample (solution by Wes Sherrod)

0 500 1000 150010-15

10-10

10-5

100Iteration balance

iteration [ ]

sum

of t

otal

hea

d [m

]

0 10 20 30

-5

0

5

10

15

Total head distribution

x-coordinate [m]

y-co

ordi

nate

[m]

0 10 20 30

-5

0

5

10

15

Elevation head distribution

x-coordinate [m]

y-co

ordi

nate

[m]

0 10 20 30

-5

0

5

10

15

Pressure head distribution

x-coordinate [m]

y-co

ordi

nate

[m]

Finite Differences MethodExample (solution by Wes Sherrod - GTREP)

0 5 10 15 20 25 30 35

0

5

10

15

20

Total head distribution

x-coordinate [m]

y-co

ordi

nate

[m] 11.0 m

2.0 m

14

Method of FragmentsThe fundamental assumption in the method of fragments is that the problem can be divided in segments with flow characterized by vertical equipotential lines (Harr 1990).

(after Holtz and Kovacks 1981)

Fragment 1 Fragment 2 Fragment 3

Method of FragmentsThe flow through a fragment m is computed as (Harr 1990)

Where k is the hydraulic conductivity, ∆htm is the total head drop in segment m, and Φm is the dimensionless form factor

m

tmhkqΦ∆⋅=

15

Method of FragmentsBecause the discharge in each segment must be the same, then

and the head loss in each segment is:

n

tn

2

2t

1

1t hkhkhkqQΦ∆⋅==

Φ∆⋅=

Φ∆⋅== L

∑∑∑

Φ∆

⋅=Φ

∆⋅=

ii

t

ii

iti Hk

hkQ

∑ΦΦ

⋅∆=∆

ii

itti Hh

Method of FragmentsFragment types and form factors

a

L

aL

=ΦFragment type I:

T

16



17

Flow Through Heterogeneous Media

Effective Permeability in Stratified Soils

Qq1

q2

q3

k1

k2

k3

1HikQ1Hikq1Hikq

1Hikq

hhorizeq

3h33

2h22

1h11

⋅⋅⋅=≡⎪⎭

⎪⎬

⎫

⋅⋅⋅=⋅⋅⋅=⋅⋅⋅=

−

H1

H2

H3

H

( )332211horizeq HkHkHkH1k ⋅+⋅+⋅⋅=−



33h22h11hh HiHiHiHi ⋅+⋅+⋅=⋅Q

k1

k2

k3

H1

H2

H3

H

Q

AikAikAik

AikQ

3h32h21h1

hverteq

⋅⋅=⋅⋅=⋅⋅=

⋅⋅= −

33

22

11verteq

HAk

QHAk

QHAk

QHAk

Q⋅

⋅+⋅

⋅+⋅

⋅=⋅

⋅−

33

22

11

kH

kH

kHverteq

Hk++

=−

18



33h22h11hh HiHiHiHi ⋅+⋅+⋅=⋅Q

k1

k2

k3

H1

H2

H3

H

Q

AikAikAik

AikQ

3h32h21h1

hverteq

⋅⋅=⋅⋅=⋅⋅=

⋅⋅= −

3AkQ

2AkQ

1AkQ

AkQ HHHH

321verteq⋅+⋅+⋅=⋅ ⋅⋅⋅⋅−

33

22

11

kH

kH

kHverteq

Hk++

=−


Zones of different hydraulic conductivities

( ) ( )1blhk1b

lhkq 2

221

11 ⋅⋅

∆⋅=⋅⋅

∆⋅=∆

1

1

2

2

2

1bl

lb

kk

⋅=

(Das 1983)

19



( ) ( )( ) ( )( ) ( )( ) ( )222

111

222

111

sinACcosACbsinACcosACbcosABsinABl

cosABsinABl

α⋅=θ⋅=α⋅=θ⋅=α⋅=θ⋅=α⋅=θ⋅=

then( )( )

( )( ) ( )

( )111

1

1

1

1

1 tantan

1cossin

sincos

lb

α=θ

=αα

=θθ

=

( )( )

( )( ) ( )

( )222

2

2

2

2

2 tantan

1cossin

sincos

lb

α=θ

=αα

=θθ

=



( )( )

( )( )1

2

2

1

2

1tantan

tantan

kk

αα

=θθ

=

Examples:

(Das 1983)

20



Flow Net Example:

(Das 1983)

Flow Through Anisotropic Media

Anisotropic Media kx ≠ ky

Laplace’s equation: 2T

2

2T

2

yh

yxh

x kk0∂

∂

∂

∂ ⋅+⋅=

where: xx'xxy

kk⋅=⋅α=

The equation may be easily solved by transforming coordinates fromThe (x,y) space to the (x’,y) space

and: ⎟⎠⎞

⎜⎝⎛α

∂=∂⇒α

='xx'xx

2T

22

2T

2TT

'xh

xh

'xh

xh

∂∂⋅α=

∂∂

⇒∂∂⋅α=

∂∂

21


Replacing into Laplace’s equation:

2T

2

2T

2

xy

2T

2

2T

2

yh

y'xh

kk

x

yh

y'xh2

x

kk

kk0

∂

∂

∂

∂

∂

∂

∂

∂

⋅+⋅⎟⎠⎞⎜

⎝⎛⋅=

⋅+⋅α⋅=

And that is: 2T

2

2T

2

yh

'xh0

∂

∂

∂

∂ +=

The problem may then be solved using flow nets by redrawing the cross-section using x’=x⋅√(ky/kx)…




The flow in an anisotropic media: y

hyy

heq

TT kxkxQ ∂∂

∂∂ ⋅⋅=⋅⋅⋅α=

yxeq kkk ⋅=

yeqy

yyeq kk

kk

kk =⋅⇒=⋅α

Transformed permeability:

Q

x

y ky

Q

α⋅x

y ky

22

Coupled FlowCoupled Phenomena

It is justify by: Le Chaterlier’s Principle & Energy Conservation

Example: Electrosmosis

(Das 1983)

Seepage Control - Filters

Seepage: Cut-off wallsImpervious blankets

Erosion and piping: FiltersRequirements

Piping: D15(filter) =< 5 D85 (soil)Permeability: D15(filter) >= 5 D15 (soil)Uniformity: D50(filter) =< 25 D50 (soil)

23

BibliographyCoduto, D. (1999). Geotechnical Engineering. Principles and Practice. Prentice-Hall. Craig, R. F. (1987). Soil Mechanics. Chapman & Hall.Das, B. (1983). Advanced Soil Mechanics.Encyclopædia Britannica (2000). http://www.encyclopedia.com.Frost, J. D. (1996). Earth Retaining Structures Class Notes. Georgia Institute of Technology. Atlanta, Georgia.Harr, M. (1990) Groundwater and Seepage. Dover PublicationsHoltz, R. and Kovaks, D. (1981). An Introduction to Geotechnical Engineering. Prentice-Hall. Matyas, E. (1994). Earth Retaining Structures Class Notes. University of Waterloo. Waterloo, Canada. McCarthy, D. (1998). Essential of Soil Mechanics and Foundation. Prentice-Hall.Powrie, W. (1997). Soils Mechanics. E&FN SPON.Santamarina, J. C. (1999). Geotechnical Engineering I Class Notes. Georgia Institute of Technology. Atlanta, Georgia.

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Table of Content - LSU Hurricane Engineeringhurricaneengineering.lsu.edu/CourseMat/CE_4780_-_Seepage.pdf5 Flow Nets The procedure consists on drawing a set of perpendicular lines: