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14.333 GEOTECHNICAL LABORATORYFlow Nets
LAPLACE'S EQUATION OF CONTINUITYSteady-State
Flow around an impervious
Sheet Pile Wall
Consider water flow at Point A:
vx = Discharge Velocity in x Direction
vz = Discharge Velocity in z Direction
Y Direction Out Of PlaneFigure 5.11. Das FGE (2005).
x
z
y
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14.333 GEOTECHNICAL LABORATORYFlow Nets
dxdydzzvv
dzdydxxv
v
zz
xx
Consider water flow at Point A(Soil Block at Pt A shown left)
Rate of water flow into soil block in x direction:
vxdzdyRate of water flow into soil block in z direction:
vzdxdy
Figure 5.11. Das FGE (2005).
Rate of water flow out of soil block in x,z directions:
LAPLACE'S EQUATION OF CONTINUITY
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14.333 GEOTECHNICAL LABORATORYFlow Nets
0
0
zv
xv
ordxdyvdzdyv
dxdydzzvvdzdydx
xv
v
zx
zx
zz
xx
Consider water flow at Point A(Soil Block at Pt A shown left)
Figure 5.11. Das FGE (2005).
Total Inflow = Total Outflow
LAPLACE'S EQUATION OF CONTINUITY
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14.333 GEOTECHNICAL LABORATORYFlow Nets
02
2
2
2
zhk
xhk
zhkikv
xhkikv
zx
zzzz
xxxx
Consider water flow at Point A(Soil Block at Pt A shown left)
Figure 5.11. Das FGE (2005).
Using Darcy’s Law (v=ki)
LAPLACE'S EQUATION OF CONTINUITY
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14.333 GEOTECHNICAL LABORATORYFlow Nets
FLOW NETS: DEFINITION OF TERMSFlow 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!
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14.333 GEOTECHNICAL LABORATORYFlow Nets
Figure 5.12a. Das FGE (2005).
FLOW NETSFLOW AROUND
SHEET PILE WALL
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14.333 GEOTECHNICAL LABORATORYFlow Nets
Figure 5.12b. Das FGE (2005).
FLOW NETSFLOW AROUND
SHEET PILE WALL
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14.333 GEOTECHNICAL LABORATORYFlow Nets
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.
FLOW NETS: BOUNDARY CONDITIONS
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14.333 GEOTECHNICAL LABORATORYFlow Nets
Figure 5.13. Das FGE (2005).
FLOW NETSFLOW UNDER ANIMPERMEABLE
DAM
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14.333 GEOTECHNICAL LABORATORYFlow Nets
nqqqq ...321
Figure 5.14. Das FGE (2005).
q k h1 h2
l1
l1 k h2 h3
l2
l2 k h3 h4
l3
l3 ...
h1 h2 h2 h3 h3 h4 ... HNd
Rate of Seepage ThroughFlow Channel (per unit length):
Using Darcy’s Law (q=vA=kiA)
Potential DropWhere:
H = Head DifferenceNd = Number of Potential Drops
FLOW NETS: DEFINITION OF TERMS
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14.333 GEOTECHNICAL LABORATORYFlow Nets
FLOW NETS: RULES FOR CREATINGFLOW 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)!
2. Equipotential lines must match known boundary conditions.
3. Flow lines can never cross.
Flow Line
Equi.
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14.333 GEOTECHNICAL LABORATORYFlow Nets
FLOW NETS: RULES FOR CREATINGFLOW NETS (FROM UTEXAS)
4. Refraction of flow lines must account for differences in hydraulic conductivity.
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.
Flow Line
Equi.
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14.333 GEOTECHNICAL LABORATORYFlow Nets
FLOW NETS: RULES FOR CREATINGFLOW NETS (FROM UTEXAS)
6. The quantity of flow between any two adjacent flow lines must be equal.
7. The quantity of flow between any two stream lines is always constant.
Flow Line
Equi.
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14.333 GEOTECHNICAL LABORATORYFlow 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.
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14.333 GEOTECHNICAL LABORATORYFlow 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.
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14.333 GEOTECHNICAL LABORATORYFlow Nets
FLOW NETS: EXAMPLES
Unconfined groundwater flow nets on a slope
Wrong Wrong
Correct!
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14.333 GEOTECHNICAL LABORATORYFlow Nets
FLOW NETS: EXAMPLES
Cross-sectional flow net of a homogeneous and isotropic aquifer (Hubbert, 1940).
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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).
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14.333 GEOTECHNICAL LABORATORYFlow Nets
FLOW NETS: DAM EXAMPLES
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14.333 GEOTECHNICAL LABORATORYFlow Nets
Figure 5.12b. Das FGE (2005).
d
f
NHN
kq
If Number of Flow Channels = Nf, then the total flow for all channels per unit length is:
Therefore, flow through one channel is:
dNHkq
FLOW NETSFLOW AROUND SHEET PILE WALL EXAMPLE
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14.333 GEOTECHNICAL LABORATORYFlow Nets
GIVEN:
Flow Net in Figure 5.17.Nf = 3Nd = 6kx=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.
Figure 5.17. Das FGE (2005).
FLOW NETSFLOW AROUND SHEET PILE WALL EXAMPLE
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14.333 GEOTECHNICAL LABORATORYFlow Nets
mmm 56.06
)67.15(
SOLUTION:
Potential Drop = dN
H
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
Figure 5.17. Das FGE (2005).
FLOW NETSFLOW AROUND SHEET PILE WALL EXAMPLE
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FLOW NETSFLOW AROUND SHEET PILE WALL EXAMPLE
SOLUTION:
dNHkq
k = 5x10-3 cm/seck = 5x10-5 m/sec
q = (5x10-5 m/sec)(0.56m)q = 2.8x10-5 m3/sec/m
fd
f qNN
HNkq
q = (2.8x10-5 m3/sec/m) * 3q = 8.4x10-5 m3/sec/m
Figure 5.17. Das FGE (2005).