1

Flow in Earth Dams

Need for Flow InformationEstimate seepage through damEstimate seepage through foundation soilsDesign of cutoffs and ground treatment programsPorewater pressure to determine up and downstream slope stabilityPrevention of piping failures

2

Piping Failures

3≥=m

c

iiFS

waterci γ

γ '=

im= mobilized hydraulic gradient

Determine im from flow net

Anisotropic K

02

2

2

2

=∂∂

+∂∂

zx kzhk

xh

0

2

2

2

2

=∂∂

+∂∂

=

zh

xh

kkif zx

3

Anisotropic kx>kz

02

2

2

2

=∂∂

+∂∂

zx kzhk

xh

02

2

2

2

=∂∂

+∂⎟⎟

⎠

⎞⎜⎜⎝

⎛∂

zh

xkk

h

x

z

x

zt k

kxx = if

Anisotropic K

02

2

2

2

=∂∂

+∂∂

zh

xh

t

zxx

zx

tx

kkkkkk

xhk

xhkv

==

∂∂

=∂∂

−=

'

'

4

Anisotropic K

Transform section with highest K

Flow Net ConstructionUnconfined Flow System

Phreatic line unknown (air-water interface)

Impermeable foundation

Phreatic surface unknown

A

B

CD

AB –equipotential line AD – flow line

BC is not equipotential or flow line

5

Unconfined flowTwo basic methods to determine phreatic

surface:

1) Draw trial flow net See Cedergren Seepage Drainage and Flownets – posted on web.

2) Numerical solution based on parabola

Numerical SolutionDupuit Solution

Modified Laplace equationAssumes:

a) Flow lines to be nearly horizontalb) Hydraulic gradient of the flow is equal to

the slope of the phreatic surface

xyi

∂∂

=

6

Dupuit Solution

y

y

L

h1h2

Impermeable foundation qf=0

x

Dupuit Solution

( )Lxhhhy 2

22

12

1 −−=

Top phreatic surface defined by

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

Lhhkq

2

22

21

7

Dupuit SolutionDoes not take into account:

Slope geometryEntrance or exit conditions

(seepage surface is missing)

Casagrande Entrance ConditionMove parabola by 0.3Δ and draw parabola from D0 to C

y

y

L

h1h2

Impermeable foundation qf=0

x

Δ

D0

0.3Δ

C

8

Exit ConditionDifferent Methods Available:

Graphical solutions:1. Schaffemak and Van Iterson2. Casagrande3. Pavovsky’s solution

Pavovsky’s Solutiony

hw

h0

a0y0

hd

hd-y0

h

a1

I II

IIIImpermeable foundation qf=0

b

β α

qI = qII = qIII

9

Pavovsky’s Solution

where m1= cotanα, m = cotanβ

hhh

mhh

ma

d

dw

−⎟⎠⎞

⎜⎝⎛ −

= ln1

0

⎟⎟⎠

⎞⎜⎜⎝

⎛−+−+= 22

110 hh

mbh

mba dd

Use trial and error to solve non-liner equations

Assume different values of h and calculate a0

Pavovsky’s Solution

a0

hh

10

Dam with Horizontal DrainDupuit Solution

y

y0

d

h

Impermeable foundation qf=0

Δ

D0

0.3Δ

a0

Horizontal drain

Dam with Horizontal Drain

d- 220 hdy +=

20

0ya =

d- 220 hdkkyq +==

11

Horizontal DrainsGood Practice Rule

Length of horizontal drain from the down stream slope should be at least H/3 where H is the height of the dam

Dams with Triangular Toe Drainy

y0

d

h

Impermeable foundation qf=0

Δ

D0

0.3Δ

a0

Toe drain

Assume y0 and a0. Error on safe side so that design phreaticsurface will be higher than expected…this will increase porewaterpressure and lower factor of safety.

Casagrande proposed corrections for toe drains – this can be ignored

12

Dams on Pervious Foundation

y

q1 k0

d

Hhs

Pervious foundation q2, Kp

D0

0.3Δ

C

impervious foundation qf=0

D b1

See Canal and River Levees by Pavol Peters Elsevier publishing 1982

Dams on Pervious Foundation

21 qqq +=

( )nb

HDkd

hHkq ps

1

220

2+

−=

1.30

3

1.44

2

1.871.231.181.15n

14520B1/D

13

Dams on Pervious FoundationAnalysis assumes

k = k0=kp

K in dam ~k in foundation