14.333 GEOTECHNICAL LABORATORY
Flow Nets
LAPLACE'S EQUATION OF CONTINUITY
Steady-State
Flow around an
impervious
Sheet Pile Wall
z
y
x
Consider water flow
at Point A:
vx = Discharge Velocity
in x Direction
vz = Discharge Velocity
in z Direction
Figure 5.11. Das FGE (2005).
Y Direction Out Of Plane
Revised 03/2013
Slide 1 of 23
14.333 GEOTECHNICAL LABORATORY
Flow Nets
LAPLACE'S EQUATION OF CONTINUITY
Consider water flow at Point A
(Soil Block at Pt A shown left)
Rate of water flow into soil block in
x direction:
vxdzdy
Rate of water flow into soil block in
z direction:
vzdxdy
Rate of water flow out of soil block
in x,z directions:
Figure 5.11. Das FGE (2005).
Revised 03/2013
v x 

dx dzdy
 vx 
x


v z 

v

dz dxdy
 z
z

 Slide 2 of 23
14.333 GEOTECHNICAL LABORATORY
Flow Nets
LAPLACE'S EQUATION OF CONTINUITY
Consider water flow at Point A
(Soil Block at Pt A shown left)
Total Inflow = Total Outflow


v x 
v z 

dx dzdy   v z 
dz dxdy  
 v x 
x
z





v x dzdy  v z dxdy  0
Figure 5.11. Das FGE (2005).
Revised 03/2013
or
v x v z

0
x
z
Slide 3 of 23
14.333 GEOTECHNICAL LABORATORY
Flow Nets
LAPLACE'S EQUATION OF CONTINUITY
Consider water flow at Point A
(Soil Block at Pt A shown left)
Using Darcy’s Law (v=ki)
 h 
vx  k xix  k x   
 x 
 h 
v z  k z iz  k z   
 z 

Figure 5.11. Das FGE (2005).
kx
Revised 03/2013
 2h
x
2
 kz
 2h
z
2
0
Slide 4 of 23
14.333 GEOTECHNICAL LABORATORY
Flow Nets
FLOW NETS: DEFINITION OF TERMS
Flow Net: Graphical Construction used to calculate
groundwater flow through soil. Comprised of Flow
Lines and Equipotential Lines.
Flow Line: A line along which a water particle moves
through a permeable soil medium. (a.k.a. streamline).
Flow Channel: Strip between any two adjacent Flow
Lines.
Equipotential Lines: A line along which the potential
head at all points is equal.
NOTE: Flow Lines and Equipotential Lines must meet at right angles!
Revised 03/2013
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14.333 GEOTECHNICAL LABORATORY
Flow Nets
FLOW NETS
FLOW AROUND
SHEET PILE WALL
Figure 5.12a. Das FGE (2005).
Revised 03/2013
Slide 6 of 23
14.333 GEOTECHNICAL LABORATORY
Flow Nets
FLOW NETS
FLOW AROUND
SHEET PILE WALL
Figure 5.12b. Das FGE (2005).
Revised 03/2013
Slide 7 of 23
14.333 GEOTECHNICAL LABORATORY
Flow Nets
FLOW NETS: BOUNDARY CONDITIONS
1. The upstream and downstream surfaces of the
permeable layer (i.e. lines ab and de in Figure 12b
Das FGE (2005)) are equipotential lines.
2. Because ab and de are equipotential lines, all the
flow lines intersect them at right angles.
3. The boundary of the impervious layer (i.e. line fg in
Figure 12b Das FGE (2005)) is a flow line, as is the
surface of the impervious sheet pile (i.e. line acd in
Figure 12b Das FGE (2005)).
4. The equipontential lines intersect acd and fg
(Figure 12b Das FGE (2005)) at right angles.
Revised 03/2013
Slide 8 of 23
14.333 GEOTECHNICAL LABORATORY
Flow Nets
FLOW NETS
FLOW UNDER AN
IMPERMEABLE
DAM
Revised 03/2013
Figure 5.13. Das FGE (2005).
Slide 9 of 23
14.333 GEOTECHNICAL LABORATORY
Flow Nets
FLOW NETS: DEFINITION OF TERMS
Rate of Seepage Through
Flow Channel (per unit length):
q1  q 2  q3  ...  q n
Using Darcy’s Law
(q=vA=kiA)
 h3  h4 
 h1  h2 
 h2  h3 
q  k 
 l3  ...
 l1  k 
 l2  k 
 l1 
 l2 
 l3 
Figure 5.14. Das FGE (2005).
Potential Drop
h1  h2  h2  h3  h3  h4  ... 
Revised 03/2013
H
Nd
Where:
H = Head Difference
Nd = Number of Potential Drops
Slide 10 of 23
14.333 GEOTECHNICAL LABORATORY
Flow Nets
FLOW NETS: RULES FOR CREATING
FLOW NETS (FROM UTEXAS)
1. Head drops between adjacent
equipotential lines must be constant
(or, in those rare cases where this is
not desirable, clearly stated, just as in
topographic contour maps)!
Equi.
2. Equipotential lines must match known
boundary conditions.
Flow Line
3. Flow lines can never cross.
Revised 03/2013
Slide 11 of 23
14.333 GEOTECHNICAL LABORATORY
Flow Nets
FLOW NETS: RULES FOR CREATING
FLOW NETS (FROM UTEXAS)
4. Refraction of flow lines must account
for differences in hydraulic
conductivity.
Equi.
5. For isotropic media (what you have).
a) Flow lines must intersect equipotential
lines at right angles.
b) The flow line-equipotential polygons
should approach curvilinear squares, as
shown in the Figure to the right.
Revised 03/2013
Flow Line
Slide 12 of 23
14.333 GEOTECHNICAL LABORATORY
Flow Nets
FLOW NETS: RULES FOR CREATING
FLOW NETS (FROM UTEXAS)
6. The quantity of flow between any two
adjacent flow lines must be equal.
Equi.
7. The quantity of flow between any two
stream lines is always constant.
Flow Line
Revised 03/2013
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14.333 GEOTECHNICAL LABORATORY
Flow Nets
FLOW NETS: DRAWING PROCEDURE
(AFTER HARR (1962, P. 23)
1. Draw the boundaries of the flow region to scale so that all equipotential
lines and flow lines that are drawn can be terminated on these
boundaries.
2. Sketch lightly three or four flow lines, keeping in mind that they are
only a few of the infinite number of curves that must provide a smooth
transition between the boundary flow lines. As an aid in spacing of
these lines, it should be noted that the distance between adjacent flow
lines increases in the direction of the larger radius of curvature.
3. Sketch the equipotential lines, bearing in mind that they must intersect
all flow lines, including the boundary streamlines, at right angles and
that the enclosed figures must be (curvilinear) squares.
Revised 03/2013
Slide 14 of 23
14.333 GEOTECHNICAL LABORATORY
Flow Nets
FLOW NETS: DRAWING PROCEDURE
(FROM HARR (1962, P. 23)
4. Adjust the locations of the flow lines and the equipotential lines to
satisfy the requirements of step 3. This is a trail-and-error process
with the amount of correction being dependent upon the position of the
initial flow lines. The speed with which a successful flow net can be
drawn is highly contingent on the experience and judgment of the
individual. A beginner will find the suggestions in Casagrande (1940)
to be of assistance.
5. As a final check on the accuracy of the flow net, draw the diagonals of
the squares. These should also form smooth curves that intersect each
other at right angles.
Revised 03/2013
Slide 15 of 23
14.333 GEOTECHNICAL LABORATORY
Flow Nets
FLOW NETS: EXAMPLES
Wrong
Wrong
Correct!
Unconfined groundwater flow nets on a slope
Revised 03/2013
Slide 16 of 23
14.333 GEOTECHNICAL LABORATORY
Flow Nets
FLOW NETS: EXAMPLES
Cross-sectional flow net of a homogeneous and isotropic aquifer (Hubbert, 1940).
Revised 03/2013
Slide 17 of 23
14.333 GEOTECHNICAL LABORATORY
Flow Nets
FLOW NETS: EXAMPLES
Contour map of the piezometric surface near Savannah, Georgia, 1957, showing closed contours
resulting from heavy local groundwater pumping (from Bedient, after USGS Water-Supply Paper 1611).
Revised 03/2013
Slide 18 of 23
14.333 GEOTECHNICAL LABORATORY
Flow Nets
FLOW NETS: DAM EXAMPLES
Revised 03/2013
Slide 19 of 23
14.333 GEOTECHNICAL LABORATORY
Flow Nets
FLOW NETS
FLOW AROUND SHEET PILE WALL EXAMPLE
Therefore, flow
through one channel
is:
H
q  k
Nd
If Number of Flow
Channels = Nf, then
the total flow for all
channels per unit
length is:
qk
Revised 03/2013
HN f
Nd
Figure 5.12b. Das FGE (2005).
Slide 20 of 23
14.333 GEOTECHNICAL LABORATORY
Flow Nets
FLOW NETS
FLOW AROUND SHEET PILE WALL EXAMPLE
GIVEN:
Flow Net in Figure 5.17.
Nf = 3
Nd = 6
kx=kz=5x10-3 cm/sec
DETERMINE:
a. How high water will
rise in piezometers at
points a, b, c, and d.
b. Rate of seepage
through flow channel
II.
c. Total rate of seepage.
Revised 03/2013
Figure 5.17. Das FGE (2005).
Slide 21 of 23
14.333 GEOTECHNICAL LABORATORY
Flow Nets
FLOW NETS
FLOW AROUND SHEET PILE WALL EXAMPLE
SOLUTION:
H
Potential Drop =
Nd
(5m  1.67 m)
 0.56m
6
At Pt a:
Water in standpipe =
(5m – 1x0.56m) = 4.44m
At Pt b:
Water in standpipe =
(5m – 2x0.56m) = 3.88m
At Pts c and d:
Water in standpipe =
(5m – 5x0.56m) = 2.20m
Revised 03/2013
Figure 5.17. Das FGE (2005).
Slide 22 of 23
14.333 GEOTECHNICAL LABORATORY
Flow Nets
FLOW NETS
FLOW AROUND SHEET PILE WALL EXAMPLE
SOLUTION:
H
q  k
Nd
k = 5x10-3 cm/sec
k = 5x10-5 m/sec
q = (5x10-5 m/sec)(0.56m)
q = 2.8x10-5 m3/sec/m
qk
HN f
Nd
 qN f
q = (2.8x10-5 m3/sec/m) * 3
q = 8.4x10-5 m3/sec/m
Revised 03/2013
Figure 5.17. Das FGE (2005).
Slide 23 of 23