flow nets.ppt

Flow Nets

Flow through a Dam

Drainageblanket

Phreatic line

UnsaturatedSoil

Flow of water

2

2

2

20

h

x

h

z

z

x

Graphical representation of solution

1. Equipotentials Lines of constant head, h(x,z)

Equipotential (EP)

Phreatic line

Flow line (FL)

2. Flow lines Paths followed by water particles - tangential to flow

Graphical representation of solution

Equipotential (EP)

Properties of Equipotentials

h(x,z) = constant (1a)

Flow line (FL)

Equipotential (EP)


h

xdx

h

zdz 0Thus: (1b)


Flow line (FL)

Equipotential (EP)


h

xdx

h

zdz 0Thus: (1b)

Equipotenial slopedz

dx

h x

h zEP

/

/(1c)


Flow line (FL)

Equipotential (EP)

zx

Geometry

vzvx

Kinematics

Properties of Flow Lines

From the geometry (2b)dx

dz

v

vFL

x

z

Flow line (FL)

Equipotential (EP)

zx

Geometry

vzvx

Kinematics


From the geometry (2b)

Now from Darcy’s law

dx

dz

v

vFL

x

z

v kh

xx

v kh

zz

Flow line (FL)

Equipotential (EP)

zx

Geometry

vzvx

Kinematics


From the geometry (2b)

Now from Darcy’s law

Hence (2c)

dx

dz

v

vFL

x

z

v kh

xx

dx

dz

h x

h zFL

v kh

zz

Flow line (FL)

Equipotential (EP)

Orthogonality of flow and equipotential lines

dz

dx

h x

h zEP

/

/

dx

dz

h x

h zFL

On an equipotential

On a flow line

Flow line (FL)

Equipotential (EP)

Orthogonality of flow and equipotential lines

dz

dx

h x

h zEP

/

/

dx

dz

h x

h zFL

On an equipotential

On a flow line

Hencedx

dz

dx

dzFL EP

1 (3)

Flow line (FL)

Equipotential (EP)

Q

X

y

z

t

T

Y

Z

X

FL

FL

Geometric properties of flow nets

Q

hh+h

h+2h

EP

Q

X

y

z

t

T

Y

Z

X

FL

FL

vQ

yx

(4a)

From the definition of flow


Q

hh+h

h+2h

EP

Q

X

y

z

t

T

Y

Z

X

FL

FL

vQ

yx

v kh

zt

(4a)

(4b)


From Darcy’s law


Q

hh+h

h+2h

EP

Q

X

y

z

t

T

Y

Z

X

FL

FL

vQ

yx

v kh

zt

Q

k h

yx

zt

(4a)

(4b)

(4c)


From Darcy’s law

Combining (4a)&(4b)


Q

hh+h

h+2h

EP

Q

X

y

z

t

T

Y

Z

X

FL

FL

vQ

yx

v kh

zt

Q

k h

yx

zt

Q

k h

YX

ZT

(4a)

(4b)

(4c)

(4d)


From Darcy’s law

Combining (4a)&(4b)

Similarly


Q

hh+h

h+2h

EP

Q

X

y

z

t

T

Y

Z

X

FL

FL

vQ

yx

v kh

zt

Q

k h

yx

zt

Q

k h

YX

ZT

(4a)

(4b)

(4c)

(4d)


From Darcy’s law

Combining (4a)&(4b)

Similarly


Q

hh+h

h+2h

EP

Conclusion

yx

zt

YX

ZT (5)

Q

a

b

c

d

D

B

C

A

h

h h


FL

Q

EP( h )

EP ( h + h )

vQ

cd

(6a)


Q

a

b

c

d

D

B

C

A

h

h h


FL

Q

EP( h )

EP ( h + h )

vQ

cd

v kh

ab

(6a)

(6b)


From Darcy’s lawQ

a

b

c

d

D

B

C

A

h

h h


FL

Q

EP( h )

EP ( h + h )

vQ

cd

Q

k h

cd

ab

v kh

ab

Q

k h

CD

AB

(6a)

(6b)

(6c)

(6d)


From Darcy’s law

Similarly

Combining (6a)&(6b)

Q

a

b

c

d

D

B

C

A

h

h h


FL

Q

EP( h )

EP ( h + h )

vQ

cd

Q

k h

cd

ab

v kh

ab

Q

k h

CD

AB

(6a)

(6b)

(6c)

(6d)


From Darcy’s law

Similarly

Combining (6a)&(6b)

Conclusioncd

ab

CD

AB

Q

a

b

c

d

D

B

C

A

h

h h


FL

Q

EP( h )

EP ( h + h )

• When drawing flow nets by hand it is most convenient to draw them so that

• Each flow tube carries the same flow Q• The head drop between adjacent EPs, h, is the

same

• Then the flow net is comprised of “SQUARES”



Demonstration of ‘square’ rectangles with inscribed circles

Drawing Flow Nets

To calculate the flow and pore pressures in the ground a flow net must be drawn.

The flow net must be comprised of a family of orthogonal lines (preferably defining a square mesh) that also satisfy the boundary conditions.

Water

Datum

H-z

z

H

(7)

Common boundary conditionsa. Submerged soil boundary - Equipotential

hu

zw

w

Water

Datum

H-z

z

H

(7)


hu

z

now

u H z

w

w

w w

( )

Water

Datum

H-z

z

H

(7)


hu

z

now

u H z

so

hH z

z H

w

w

w w

w

w

( )

( )

Permeable Soil

Flow Linevn=0

vt

Impermeable Material

Common boundary conditionsb. Impermeable soil boundary - Flow Line

Common boundary conditionsc. Line of constant pore pressure - eg. phreatic surface

hu

zw

w

Head is given by


hu

zw

w

hu

zw

w

Head is given by

and thus


hu

zw

w

hu

zw

w

uw

0

Head is given by

and thus

now if pore pressure is constant


hu

zw

w

hu

zw

w

h z

uw

0

Head is given by

and thus

now if pore pressure is constant

and hence (8)

Procedure for drawing flow nets

• Mark all boundary conditions

• Draw a coarse net which is consistent with the boundary conditions and which has orthogonal equipotentials and flow lines. (It is usually easier to visualise the pattern of flow so start by drawing the flow lines).

• Modify the mesh so that it meets the conditions outlined above and so that rectangles between adjacent flow lines and equipotentials are square.

• Refine the flow net by repeating the previous step.

Value of head on equipotentials

Phreatic line

hH

Numberofpotentialdrops (9)

Datum

15 m

h = 15m

h = 12m h = 9m h = 6mh = 3m

h = 0

For a single Flow tube of width 1m: Q = k h (10a)

Calculation of flowPhreatic line

15 m

h = 15m

h =12m h = 9m h = 6mh = 3m

h = 0


For k = 10-5 m/s and a width of 1m Q = 10-5 x 3 m3/sec/m (10b)


15 m

h = 15m

h =12m h = 9m h = 6mh = 3m

h = 0



For 5 such flow tubes Q = 5 x 10-5 x 3 m3/sec/m (10c)


15 m

h = 15m

h =12m h = 9m h = 6mh = 3m

h = 0




For a 25m wide dam Q = 25 x 5 x 10-5 x 3 m3/sec (10d)


15 m

h = 15m

h =12m h = 9m h = 6mh = 3m

h = 0




For a 25m wide dam Q = 25 x 5 x 10-5 x 3 m3/sec (10d)


15 m

h = 15m

h =12m h = 9m h = 6mh = 3m

h = 0

Q kH

NN

hfNote that per metre width (10e)

P5m

hu

zw

w

(11a)

Calculation of pore pressurePhreatic line

P5m

Pore pressure from

15 m

h = 15m

h = 12m h = 9m h = 6mh = 3m

h = 0

P5m

hu

zw

w

(11a)

uw w [ ( )]12 5 (11b)

Calculation of pore pressurePhreatic line

P5m

Pore pressure from

At P, using dam base as datum

15 m

h = 15m

h = 12m h = 9m h = 6mh = 3m

h = 0

StrandedVessel

WaterSupply

SoftSeaBottom

WellPoint

ReactionPile

Example Calculating Pore Pressures

20 m

10 m

Step 1: Choose a convenient datum. In this example the sea floor has been chosen

Then H1 = 40 mH2 = 1 m.

The increment of head, h = 39/9 = 4.333 m

A B C D E

Step 2: Calculate the head at points along the base of the vessel. For convenience these are chosen to be where the EPs meet the vessel (B to E) and at the vessel centerline (A). Hence calculate the pore water pressures.

At B Head = H1 - 5 h = H2 + 4 h = 18.33 m

Pore pressure at B = 18.33 w = 179.8 kPaww zhu )]([

Step 3: Calculate the upthrust (Force/m) due to pore pressures

7.02

3.529.948.1

2

9.943.1375.2

2

3.1378.1795

2

8.1791.2012

= 3218 kN/m

Without pumping Upthrust = 20 1 9.81 = 196 kN/m

Upthrust due to Pumping = 3218 – 196 = 3022 kN/m

0

50

100

150

200

250

0 1 2 3 4 5 6 7 8 9 10

Distance from centreline (m)

Po

re W

ate

r P

res

su

re (

kP

a)

Flow required, h

f

N

NHkQ = 24 108.1

9

1439103 m3/m/sec

flow nets.ppt

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