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Flow Nets

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k hx

k hz

H V

2

2

2

2 0 For a homogeneous soil

For an isotropic soil

2

2

2

2 0

h

x

h

z

Laplace Equation

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Laplace Equation -1 D solution

The 1D solution of the Laplace equation the simplest

Boundary Value Problem. Now consider the case of only the

vertical flow exists, then the Laplace equation is simplified to

02

2

z h

The solution of this equation is easy to get by having a

integration of h with respect to z twice

02

2

z

h

1tant Acons

z

h

21 Az Ah

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Laplace Equation -1 D solutionThe constants A1 and A2 can be determined by the boundary

conditi ons. For soil 1

Condition 1: h= h1 at z =0

Condition 2: h= h2 at z =H1

1

1

12h z

H

h h h

10 H z for

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For soi l 2

Condition 1: h= h2 at z = H1

Condition 2: h= 0 at z =H1+ H2

2

1

2

2

21

H

H h z

H

h h

211 H H z H for

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From the continuity of flow q1 = q2 = q

AH

h k A

H

h h k q

2

2

2

1

21

1

0

or

2

2

1

1

1

11

2

H

k

H

k H

k h h

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Substituting for h2 in equation of head for soil 1

1

1

12h z

H

h h h

We obtain

1221

2

1 1

H k H k

z k h h

Now we can predict the head in soil 1 if we know the hydraulic

conductivities in soil 1 and soil 2

10 H z for

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Now for soil 2

Substituting for h2 We obtain

z H H H k H k

k h h

21

1221

1

1

Now we can predict the head in soil 2 if we know the hydraulic

conductivities in soil 1 and soil 2

211 H H z H for

2

1

2

2

21

H

H h z

H

h h

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Flow Net for One Dimensional Flow

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Seepage is defined as the flow of a fluid,usually water, through a soil under a

hydraulic gradient.

10

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concrete dam

impervious strata

soil

Stream line is simply the path of a water molecule.

datum

hL

TH = 0TH = hL

From upstream to downstream, total head steadily decreasesalong the stream line.

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Equipotential line is simply a contour of constant

total head.

concrete dam

impervious strata

soil

datum

hL

TH = 0TH = hL

TH=0.8 hL

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A network of selected stream lines and equipotential lines.

concrete dam

impervious strata

soil

curvilinearsquare

90º

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Graphical representation of solution

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

Equipotential (EP)

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Phreatic line

Flow line (FL)

2. Flow lines Paths followed by water particles -

tangential to flow

Graphical representation of solution

Equipotential (EP)

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Properties of Equipotentials

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

Flow line (FL)

Equipotential (EP)

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hx

dx hz

dz 0Thus: (1b)


Flow line (FL)

Equipotential (EP)

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hx

dx hz

dz 0Thus: (1b)

Equipotenial sloped z

d x

h x

h zE P

/

/(1c)


Flow line (FL)

Equipotential (EP)

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z

x

Geometry

vz

vx

Kinematics

Properties of Flow Lines

From the geometry (2b)dx

dz

v

vFL

x

z

Flow line (FL)

Equipotential (EP)

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z

x

Geometry

vz

vx

Kinematics


From the geometry (2b)

Now from Darcy‟s law

dx

dz

v

vFL

x

z

v k hx

x v k h

zz

Flow line (FL)

Equipotential (EP)

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z

x

Geometry

vz

vx

Kinematics


From the geometry (2b)

Now from Darcy‟s law

Hence (2c)

dx

dz

v

vFL

x

z

v k hx

x

dx

dz

h x

h zFL

v k hz

z

Flow line (FL)

Equipotential (EP)

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Orthogonality of flow and equipotential lines

d z

d x

h x

h zE P

/

/

dxdz

h xh zFL

On an equipotential

On a flow line

Flow line (FL)

Equipotential (EP)

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Orthogonality of flow and equipotential lines

d z

d x

h x

h zE P

/

/

dxdz

h xh zFL

On an equipotential

On a flow line

Hencedx

dz

dx

dzFL EP

1 (3)

Flow line (FL)

Equipotential (EP)

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Q

X

y

z

t

T

Y

Z

X

FL

FL

Geometric properties of flow nets

Q

hh+h

h+2h

EP

G i i f fl

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Q

X

y

z

t

T

Y

Z

X

FL

FL

v

Q

yx

(4a)

From the definition of flow


Q

hh+h

h+2h

EP

G t i ti f fl t

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Q

X

y

z

t

T

Y

Z

X

FL

FL

v

Q

yx

v k h

zt

(4a)

(4b)


From Darcy‟s law


Q

hh+h

h+2h

EP

G t i ti f fl t

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Q

X

y

z

t

T

Y

Z

X

FL

FL

v

Q

yx

v k h

zt

Q

k h

yx

zt

(4a)

(4b)

(4c)


From Darcy‟s law

Combining (4a)&(4b)


Q

hh+h

h+2h

EP

G t i ti f fl t

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Q

X

y

z

t

T

Y

Z

X

FL

FL

v

Q

yx

v k h

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

G t i ti f fl t

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Q

X

y

z

t

T

Y

Z

X

FL

FL

v

Q

yx

v k h

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)

G t i ti f fl t

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Q

a

b

c

d

D

B

C

A

h

h h


FL

Q

EP( h )

EP ( h + h )

G t i ti f fl t

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v

Q

cd

(6a)


Q

a

b

c

d

D

B

C

A

h

h h


FL

Q

EP( h )

EP ( h + h )

G t i ti f fl t

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v

Q

cd

v k h

ab

(6a)

(6b)


From Darcy‟s law Q

a

b

c

d

D

B

C

A

h

h h


FL

Q

EP( h )

EP ( h + h )

G t i ti f fl t

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v

Q

cd

Q

k h

cd

ab

v k h

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 )


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v

Q

cd

Q

k h

cd

ab

v k h

ab

Q

k h

CD

AB

(6a)

(6b)

(6c)

(6d)


From Darcy‟s law

Similarly

Combining (6a)&(6b)

Conclusion

cd

ab

CD

AB

Q

a

b

c

d

D

B

C

A

h

h h


FL

Q

EP( h )

EP ( h + h )


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When drawing flow nets by hand it is most

convenient to draw them such 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”



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Demonstration of „square‟ rectangles with inscribed circles

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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 squaremesh) that also satisfy the boundary conditions.

Common boundary conditions

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Water

Datum

H-z

z

H

(7)


a. Submerged soil boundary - Equipotential

hu

zw

w


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Water

Datum

H-z

z

H

(7)



hu

z

now

u H z

w

w

w w

( )


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Water

Datum

H-z

z

H

(7)



hu

z

now

u H z

so

hH z

z H

w

w

w w

w

w

( )

( )


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Permeable Soil

Flow Linevn=0

vt

Impermeable Material


b. Impermeable soil boundary - Flow Line

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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 theflow 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

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Value of head on equipotentials

Phreatic line

h H

Number of potential drops

(9)

Datum

15 m

h = 15m

h = 12m h = 9m h = 6m

h = 3m

h = 0

Calculation of flow

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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 = 6m

h = 3m

h = 0

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Calculation of flow

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

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 = 6m

h = 3m

h = 0

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Calculation of flow

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

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

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 = 6m

h = 3m

h = 0

Q k H

N

Nh

f Note that per metre width (10e)

Calculation of pore pressure

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P5m h u 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

Calculation of pore pressure

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P5m h u zw

w

(11a)

u w 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

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