S EEPAGE AND F LOW N ETS
6.3 FLOW NET FOR TWO-DIMENSIONAL FLOW
It may be necessary to use flow nets to evaluate flow, where the directions of flow are irregu-lar, or where the flow boundaries are not well-defined. Flow nets are a pictorial method of studying the path of the moving water.
In moving between two points, water tends to travel by the shortest path. If changes in direction occur, the changes take place along smooth curved paths. Equipotential lines must cross flow lines at right-angles since they represent pressure normal to the direction of flow.
The flow lines and equipotential lines together form the flow net and are used to determine the quantities and other effects of flow through soils.
During seepage analysis, a flow net can be drawn with as many flow lines as desired.
The number of equipotential lines will be determined by the number of flow lines selected.
Generally speaking, it is preferable to use the fewest flow lines that still permit reasonable depiction of the path along the boundaries and within the soil mass. For many problems, three or four flow channels (a channel being the space between adjacent flow lines) are sufficient.
In this section the flow nets for three situations involving two-dimensional fluid flow are discussed. The first and second—flow under a sheet pile wall and flow under a concrete dam—are cases of confined flow since the boundary conditions are completely defined. The third—flow through an earth dam—is unconfined flow since the top flow line is not defined in advance of constructing the flow net. The top flow line or the phreatic line has to be deter-mined first. Thereafter, the flow net may be completed as usual.
6.3.1 Flow under Sheet Pile Wall
Figure 6.2 shows a sheet pile wall driven into a silty soil. The wall runs for a considerable length in a drirection perpendicular to the paper; thus, the flow underneath the sheet pile wall may be taken to be two-dimensional.
The boundary conditions for the flow under the sheet pile wall are; mb, upstream equipotential; jn, downstream equipotential; bej, flow line and pq, flow line. The flow net
SEEPAGE AND FLOW NETS 169 shown has been drawn within these boundaries. With the aid of flow net, we can compute the seepage under the wall, the pore pressure at any point and the hydraulic gradient at any point. A water pressure plot, such as that shown in Fig. 6.2 is useful in the structural design of the wall.
Silty soil
Flow line Equipotential line
Impervious a
l
b j n
m
q p
e
a
e e
f f g j
g j b c d e
0
(a) Flow net (b) Water pressure on the wall
l Sheet pile wall
Fig. 6.2 Sheet pile wall
6.3.2 Flow under Concrete Dam
Figures 6.3 to 6.7 show a concrete dam resting on an isotropic soil. The sections shown are actually those of the spillway portion. The upstream and tail water elevations are shown. The first one is with no cut-off walls, the second with cut-off wall at the heel as well as at the toe, the third with cut off-wall at the heel only, the fourth with cut-off wall at the toe only and the fifth is with upstream impervious blanket. The boundary flow lines and equipotentials are known in each case and the flow nets are drawn as shown within these boundaries. The effect of the cut off walls is to reduce the under seepage, the uplift pressure on the underside of the dam and also the hydraulic gradient at the exit, called the ‘exit gradient’. A flow net can be understood to be a very powerful tool in developing a design and evaluating various schemes.
Dam H
Impervious
Fig. 6.3 Concrete dam with no cut-off walls
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Impervious
H L
Dam
Fig. 6.4 Concrete dam with cut-offs at heel and at toe
Impervious Dam H
Fig. 6.5 Concrete dam with cut-off at heel
Impervious Dam H
Fig. 6.6 Concrete dam with cut-off wall at toe
SEEPAGE AND FLOW NETS 171
H Dam
Impervious blanket Filter
Impervious
Fig. 6.7 Concrete dam with impervious blanket on the upstream side and filter on the downstream side
Top flow line
Impervious
Top flow line
Impervious
Pervious blanket
Top flow line
Impervious
Chimney drain
Drain pipe
Top flow line
Impervious
Rock toe
(d) Homogeneous dam with rock toe on the downstream side (c) Homogeneous dam with chimney drain (b) Homogeneous dam with underdrain on pervious blanket
(a) Homogeneous dam without internal drain
Fig. 6.8 Top flow line for typical cases of homogeneous earth dams on impervious foundation with different internal drainage arrangements
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6.3.3 Flow through Earth Dam
The flow through an earth dam differs from the other cases in that the top flow line is not know in advance of sketching the flow net. Thus, it is a case of unconfined flow. The determination of the top flow line will be dealt with in a later section.
The top flow line as well as the flow net will be dependent upon the nature of internal drainage for the earth dam. Typical cases are shown in Fig. 6.8; the top flow line only is shown.
Assuming that the top flow line is determined, a typical flow net for an earth dam with a rock toe, resting on an impervious foundation is shown in Fig. 6.9:
B
A
Impervious D
Rock toe
C C
Fig. 6.9 Flow net for an earth dam with rock toe (for steady state seepage)
AB is known to be an equipotential and AD a flow line. BC is the top flow line; at all points of this line the pressure head is zero. Thus BC is also the ‘phreatic line’; or, on this line, the total head is equal to the elevation head. Line CD is neither an equipotential nor a flow line, but the total head equals the elevation head at all points of CD.