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Sepaage and flownets
Lecture 4
References
Braja M Das, Principal of Geotechnical Engineering, fourth edition, PWS Publishing Company, 1998, BostonMuni Budu, Soil Mechanics and Foundations, 2ndedition, John Wiley & Sons, 2007, USAR.F. Craig, Soil Mechanics, (English & terjemahan Prof.Dr.Ir Budi S Supandji)Cernica, J.N., Soil Mechanics, John Wiley & Sons,1995Holtz, R.D., Kovacs, W.D., An introduction to Geotechnical Engineering, Prentice Hall, N.J., 1981
Laplace equations
The assumptions :Darcys law is validThe soil is homogeneous and saturatedThe soil and water are imcompressibleNo volume change occurs
Laplace equations of continuity A B
Impermeable soil layerWater incompressible and no vol. change :Flow x direction
Flow z direction
Laplace equations of continuity A B
Impermeable soil layer
Darcys law : Isotrpic soil kx=kz
Simple flow S F Flow in Z direction
A1 dan A2 are constant;
For flow through layer 1 :
Simple flow S F Flow in Z direction
A1 dan A2 are constant;
For flow through layer 2 :
From condtion 2:
Simple flow S F from and
Flow of layer 1 = flow of layer2
Simple flow S F Example
H1 = 300mm, H2 = 500mm, h1=600mm at z=200mm and h=500mm
Determine h at z=600 mm
Solution Z=200mm hence
Because z=600mm is located in soil layer 2 hence,
h = 179.9 mm
Water In
)h =hA - hB
Head Loss orHead Difference or Energy Loss
hA
hB
i = Hydraulic Gradient
(q)Water
out
Datum
hA
W.T.
hB
)h = hA - hB
W.T.
Impervious Soil
Impervious Soil
ZA
Datum
ZB
q = v . A = k i A = k AhL
Three different scenarios (a) Static (b) Flow-up (c) Flow-down
Stresses due to Flow
12
Stresses due to Flow
X
soil
hw
L
Static Situation (No flow)
z
v = whw + satz
u = w (hw + z)
v ' = ' z
At X,
14 ft
3 ft
12 ft
In Flow
Out Flow 2 ft
4 ft
Datum
3 ft
3 ft
8 ft
Piezometer
A
B
C
D
u =
6 x
62.4
u =
14 x
62.
4
No Seepage
Buoyancy
Ws
Ws
Ws
Ws
Ws
Stresses due to FlowDownward Flow
hw
L
flow
X
soil
z
v = whw + satz
w hw + w(L-hL)(z/L)
v ' = ' z + wiz
At X,
hLu = w hw
u = w (hw+L-hL)
as for static case
= w hw + w(z-iz)
= w (hw+z) - wiz
Reduction due to flow
Increase due to flow
u =
10 ft
3 ft
12 ft
In Flow
Out Flow
2 ft
4 ft
Datum
3 ft
3 ft
8 ft
Piezometer
A
B
C
D
u =
6 x
62.4
-
u
u =
17 x
62.
4
Downward Seepage
Buoyancy - Seepage Force
Ws
Ws
Ws
Ws
Ws
Seepage Force
16
Stresses due to Flow
flow
Upward Flow
hw
LX
soil
z
v = whw + satz
w hw + w(L+hL)(z/L)
v ' = ' z - wiz
At X,
hL
u = w hw
u = w (hw+L+hL)
as for static case
= w hw + w(z+iz)
= w (hw+z) + wiz
Increase due to flow
Reduction due to flow
u =
17 ft
3 ft
12 ft
In Flow
Out Flow 2 ft
4 ft
Datum
3 ft
3 ft
8 ft
Piezometer
A
B
C
D
u =
6 x
62.4
+
u
u
u =
17 x
62.
4
Upward Seepage
Buoyancy + Seepage Force
Ws
Ws
Ws
Ws
Ws
18
Quick Condition in Granular SoilsDuring upward flow, at X:
v ' = ' z - wizflow
hw
LX
soil
z
hL
= izw
w '
Critical hydraulic gradient (ic)
If i > ic, the effective stresses is negative.
i.e., no inter-granular contact & thus failure.
- Quick condition
Flow nets
To construct a flownet The Equipotensial lines intersect the flow line at right angle The flow elemens formed aproximate squares
Flownet is a combination of flow lines and equipotensial lineA flow line is a line along which water particle will travel from upstream to downstream.An equipotensial line is line along which the potensial head at all points is equal
Seepage Terminology
concrete dam
impervious strata
soil
Stream/flow line is simply the path of a water molecule.
datum
hL
TH = 0TH = hL
From upstream to downstream, total head steadily decreases along the stream line.
Seepage TerminologyEquipotential line is simply a contour of constant total head.
concrete dam
impervious strata
soil
datum
hL
TH = 0TH = hL
TH=0.8 hL
FlownetA network of selected stream lines and equipotentiallines.
concrete dam
impervious strata
soil
curvilinear square
90
Equipotential Lines
Flow Element
Principles of the Flow Net
Piezometer
)h = head loss = one drop
Datum
1
2
3
4
5
Principles of the Flow Net
Equipotential LinesTotal heads along this line are the same
Flow Element
Seepage calculation from flow net 1 Square
From Darcys law v=k i A
and
where H= head difference between upstream and down streamNd = number of potential drops
Seepage calculation from flow net cont
2 Rectangular
or
Seepage calculation from flow net cont
Flow channels 1&2 have square elements
Flow channel 3 has elemenrectangular b/l=0.38
Flow nets in anisotropic soil
4.
For anisotropic soil kx kz
To construct the flow net use the following procedures:1. Adopt vertical scale (z axis) fro drawing the cross section2. Adopt horisontal scale (x axis) such that horisontal scale =
kz/kx3. Plot the vertical section through the permeable layer
parallel to the direction of flow 4. Draw the flow net for the permeable layer on the section
obtained from step 3; with flow line intersecting equipotential line at right angles and the elements as approximate squares
The rate of seepage can be calculated :
where H =total head loss,Nf dan Nd = number of flow channels and potensial
drops
Quantity of Seepage (Q)
d
fL N
NkhQ = .per unit length normal to the plane
# of flow channels
# of equipotential drops
impervious strata
concrete dam
hL
head loss from upstream to downstream
Heads at a Point X
impervious strata
concrete dam
datum
X
z
hL
TH = hL TH = 0
Total head = hL - # of drops from upstream x h
h
Elevation head = -z
Pressure head = Total head Elevation head d
L
Nh
=
Uplift pressure under hydraulic structures
Assuming kx = kz = k Nd =7 H=21ft Head loss for each potensial
drop = H/Nd=3 Uplift pressure at a =
(Pressure head at a) x w[(21+6)-3]x w =24 w
Uplift pressure at b =[27-(2)(3)] w= 21 w
Uplift pressure at f =[27-(6)(3)] w = 9 w
Weir
Uplift force under hydraulic structures
Seepage through an earth dam on impervious base
Flow through a homogeniuos earth dam constructed over impervious baseConsidering cde
Rate of seepage at the section bf
33
FiltersUsed for: facilitating drainage preventing fines from being washed away
Used in:
earth dams
retaining walls
Filter Materials:
granular soils
geotextiles
34
Granular Filter DesignTwo major criteria:
(a) Retention Criteria
(b) Permeability Criteria
- to prevent washing out of fines
- to facilitate drainage and thus avoid build-up of pore pressures
Filter grains must not be too coarse
Filter grains must not be too fine
granular filter
35
Granular Filter DesignRetention criteria:D15, filter < 5 D85,soil
- after Terzaghi & Peck (1967)
Permeability criteria:D15, filter > 4 D15,soil
average filter pore size
D15, filter < 20 D15, soil
D50, filter < 25 D50, soil- after US Navy (1971)
GSD Curves for the soil and filter must be parallel
Example
From the left figure Nf =3 Nd = 6
Head loss for each drop = h/Nd = 10/6 = 1.667
Potensial drop at a = 1x 1.667 ftWater in the piezometerAt a = 15-1.667=13.333 ft above ground surface
(ags)At b = 15-2x1.667= 11.67 ft agsAt c = 15-5x1.667= 8.67 ft agsAt d = 15-5x1.667= 8.67 ft ags
kx= kz =k = 5x 10-3 cm/sec= 1.64 x 10-4 ft/sec
q = (1.64 x 10-4) x 1.667= 2.73x10-4
ft3/sec/ft
Impermeable layer
Lecture 4Slide Number 2Laplace equationsLaplace equations of continuityLaplace equations of continuitySimple flow Simple flow Simple flow Simple flow Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Flow netsSeepage TerminologySeepage TerminologyFlownetSlide Number 23Slide Number 24Seepage calculation from flow netSeepage calculation from flow net contSeepage calculation from flow net cont Flow nets in anisotropic soilQuantity of Seepage (Q)Heads at a Point XUplift pressure under hydraulic structuresSeepage through an earth dam on impervious baseSlide Number 33Granular Filter DesignSlide Number 35Example