TOPIC 2 STEADY STATE FLOW THROUGH SOIL

Course : S0705 – Soil MechanicYear : 2008

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CONTENT

• PERMEABILITY / CAPILARITY (SESSION 7 : F2F)• SEEPAGE (SESSION 8 : F2F)• FLOW NET / JARINGAN ALIRAN (SESSION 9 – 10 : F2F)

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SESSION 7 PERMEABILITY / CAPILARITY

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INTRODUCTION

• DEFINITION :THE VELOCITY OR THE CAPABILITY OF WATER/FLUDI PASS THROUGH POROUS MEDIA

• NOTATION : k

• UNIT :m/s, cm/s

• PURPOSE :– To evaluate the seepage through dam– To evaluate the uplift force or seepage force under hydraulic structure for

stability analysis– To control seepage velocity– To determine the consolidation time rate

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– LABORATORY• CONSTANT HEAD (TINGGI KONSTAN)• FALLING HEAD (TINGGI JATUH)

– FIELD• UNCONFINED AQUIFER (AKIFER BEBAS)• CONFINED AQUIFER (AKIFER TERKEKANG )• INCONSTANT WATER HEIGHT (TINGGI AIR TIDAK TETAP)

DETERMINATION OF PERMEABILITY COEFFICIENT

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CONSTANT HEAD (TINGGI KONSTAN)

– SUITABLE FOR SANDY SOIL, SAND OR GRAVEL WHICH HAVE BIG VOID RATIO VALUE

– BASIC EQUATION :

A.h.tQ.Lk

.tLhk. A. A.(k.i).t A.v.t Q

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FALLING HEAD (TINGGI JATUH)

– MORE ECONOMICAL FOR LONG TERM TEST– BASIC EQUATION :

2

1ln..

..-

.....

-

dtdh-v

hh

tALak

LhkA

dtdhaqq

LhkAikAvAq

dtdhaq

keluarmasuk

keluar

masuk

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UNCONFINED AQUIFER (AKIFER BEBAS)

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UNCONFINED AQUIFER (AKIFER BEBAS)

2122

1

2

1

2

21

22

.

ln.

ln

.....2..

hhrrQ

k

rrhhkhr

drdhkQ

Q

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CONFINED AQUIFER (AKIFER TERKEKANG )

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CONFINED AQUIFER (AKIFER TERKEKANG )

21o

2

1

hh.h..2rrlog.Q.3,2

k

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INCONSTANT WATER HEIGHT (TINGGI AIR TIDAK TETAP)

Ly2.

rL20

ty.

yr.40

k

2r

yL

y

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RANGE OF k VALUE

210D.Ck Cm/s

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RANGE OF k VALUE

C is coefficient of soil roughness (published by Hazen), the value range from 40 to 150 i.e. as follow :

C Sand (one or all of the next characteristics)

40-80 Very fine, well graded or contain some fine particle

80-120 Rather rough, poor graded or clean, rough and well graded

120-150 Very rough, very poor graded, contain gravel and clean

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EQUIVALENT COEFFICIENT OF PERMEABILITY OF LAYERED SOIL

– Equivalent Coefficient of Vertical Permeability (kv’) – Equivalent Coefficient of Horizontal Permeability (kh’)

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Equivalent Coefficient of Vertical Permeability (kv’)

Basic Concept – qin = qout

– v constant

n

nn

2

22

1

11 H

h.k...Hh.k

Hh.ki'.kvv

vh

kH.....

vh

kH;

vh

kH;

vh

kH n

n

n3

3

32

2

21

1

1

n

n

3

3

2

2

1

1n321

kH...

kH

kH

kH

vh...

vh

vh

vh

LH...HHH n321

Lh'.kvv

n

n

3

3

2

2

1

1k

H...kH

kH

kH

L'kv

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Equivalent Coefficient of Vertical Permeability (kv’)

ikhLvAq average '...

i.H.k...i.H.ki.H.ki'.kh.L nn2211

LH.k...H.kH.k'kh nn2211

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EXAMPLE 1

Question :Find the permeability coefficient of sand in ft/min

q = 1 ft3/hr

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EXAMPLE 1

21t hhh

1

1

1 Lh.k

Aq

SECTION 1

k.AL.qh1

11

2

2

2 Lh.k

Aq

k.AL.qh2

22

k.AL.q

k.AL.qh

2

2

1

1t

SECTION 2

TOTAL

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EXAMPLE 1

k.10600.1

k.20400.120

k = 4 ft/hour = 6,67x10-2 ft/min

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EXAMPLE 2

q

Questions :- determine h- determine q in cc/sec

Section 1 Section 2

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EXAMPLE 2

Section 11111 A.i.kq

Section 2

2222 A.i.kq

25.4050.02.01

hq 25.

405.007.02

hq

21 qq )5.(007.0)50.(02.0 hh

h = 38.33 cm

Determination of h

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EXAMPLE 2

1111 A.i.kq 2222 A.i.kq

25.4033.3850.02.0

q

Determination of water flow rate

or

q = 0.15 cc/s

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SESSION 8 SEEPAGE

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INTRODUCTION

• DEFINITIONVOLUME OF WATER/FLUID FLOW IN OR FLOW OUT AT A MEDIA OR CERTAIN SOIL MASS

• PURPOSE– TO DETERMINE THE INFLUENCE OF SEEPAGE TO THE STABILITY

OF STRUCTURE OR DAM– TO ESTIMATE THE FLOW VELOCITY AND FLOW RATE AT

DEWATERING WORK

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WATER FLOW EQUATION

– BASIC OF THEORY• DARCY LAW

• BERNOULLI LAW

• CONTINUITY EQUATION

i.kv Lhi

tconsAvAvq tan.. 2211

2w

222

1w

121 z.g

g.p

g2vz.g

g.p

g2v

= constant energy

ni.k'v

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WATER FLOW EQUATION

dxdyvdxdzvdydzvq zyxlow inf

Inflow water per time unit:

Outflow water per time unit:dxdydz

zvvdxdzdy

yv

vdydzdxxvvq z

zy

yx

xoutflow

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WATER FLOW EQUATION

dtdVdxdydz

zv

yv

xv zyx

qinflow = qoutflow

te

eo11

tW1

zv

yv

xv w

w

zyx

CONTINUITY EQUATION

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WATER FLOW EQUATION

0te

0zv

yv

xv zyx

STEADY STATE CONDITION:

WATER FLOW VELOCITY :

zhkikv

yhkikv

xhkikv

zzzz

yyyy

xxxx

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WATER FLOW EQUATION

0zhk

yhk

xhk 2

2

z2

2

y2

2

x

0zhk

zyhk

yxhk

x zyx

0zh

yh

xhh 2

2

2

2

2

22

HOMOGEN SOILk constant at x,y,z direction

ISOTROPY SOILkx = ky = kz = k

LAPLACE EQUATION

0zh

xhh 2

2

2

22

TWO DIMENSION

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WATER FLOW EQUATION

• SEEPAGE SOLUTION– CLOSED FORM SOLUTION– MODEL SOLUTIONS– APPROXIMATE SOLUTIONS

• NUMERICAL SOLUTIONS• GRAPHICAL SOLUTIONS FLOW NET

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SESSION 9-10 FLOW NET / JARINGAN ALIRAN

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DEFINITION

Combination of 2 line group which perpendicular each other i.e. :– Flow Line

A bundle of line or points which defined flow direction – Equipotential Line

The position of point which have same total head

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FLOW NET / JARINGAN ALIRAN

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FLOW NET / JARINGAN ALIRAN

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FLOW NET / JARINGAN ALIRAN

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FLOW NET / JARINGAN ALIRAN

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FLOW NET / JARINGAN ALIRAN

• GUIDANCE OF FLOW NET DRAWING– The water surface of upper stream and lower stream is equipotential line– The interface line of water and soil is equipotential line– The flow line perpendicular with equipotential line– The impermeable surface of boundary line is flow line– The shape area formed by flow line and equipotential line is square

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FLOW NET / JARINGAN ALIRAN

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FLOW NET / JARINGAN ALIRAN

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FLOW NET / JARINGAN ALIRAN

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FLOW NET / JARINGAN ALIRAN

bh.k).1.a(i.k.Av.Aq

d

21

Nhhh

d

21f N

hhba.k.Nqq

21 hhH

baH.

NNkq

d

f H.NNkq

d

f a = b

h

h+h

q

a b

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EXAMPLE 3

4,50 m

8,60 m

A B C D

E

Datum

Sheet Piling

6,0 m

0,5 m

k = 1,5 x 10-6 m/s

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EXAMPLE 3

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EXAMPLE 3Nd = 12

Nf = 4,3H = 4,0 m

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EXAMPLE 3

H.NNkq

d

f

H.Nnh

d

dP

msmxxq ./1015.200.4.123,4.105.1 366

mhP 33.34.1210

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EXAMPLE 4

k = 2,5 x 10-5 m/s

5 m

15 m12 m

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EXAMPLE 4Nd = 15 Nf = 4,7

H = 4,0 m

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EXAMPLE 4

H.NNkq

d

f

msmxxq ./101.300.4.157,4.105.2 355

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SEEPAGE FORCE / GAYA REMBESAN

H

h2

L

h1

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SEEPAGE FORCE / GAYA REMBESAN

A).hh.(A.L.F 21wt

TOTAL FORCE

BODY FORCE (GAYA BADAN)

volumeForceTotal

FForceBody )(

L Soil weight = t.L.A

w . h2 . A

w . h1 . A

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SEEPAGE FORCE / GAYA REMBESAN

wbouyancy

wtwt

wt

iF

iLLHF

ALAhhALF

.

)1(

.)..(.. 21

w.i SEEPAGE BODY FORCE (j)=

CRITICAL CONDITION

e11Gi

0.i

s

w

bouyantc

wbouyant

bouyancy = t - w

H

h2

L

h1

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EXAMPLE 5

k = 1x10-3 cm/sn = 0.67

Questions :1. Water Flow Rate2. Flow Velocity3. Seepage Velocity4. Seepage Force at point A

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EXAMPLE 5• Water Flow Rate

• Flow Velocity

A.i.kq 144

LHi

A10x1A.1.10x1q 55

i.kv

s/m10x11.10x1v 55

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EXAMPLE 5• Seepage Velocity

• Seepage Force

nv

ni.k'v

smxxv /105.167.0101' 5

5

ws .iF

2/10001000.1 mkgFs