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# Lecture 10 - Plotting of Phreatic Line for Seepage through Earth Dams Notes | EduRev

## : Lecture 10 - Plotting of Phreatic Line for Seepage through Earth Dams Notes | EduRev

``` Page 1

Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                    1
Module 2
Lecture 10
Permeability and Seepage -6
Topics
1.2.11 Plotting of Phreatic Line for Seepage through Earth Dams
1.2.12 Entrance, Discharge, and Transfer Conditions of Line of Seepage
through Earth Dams
1.2.13 Flow net Construction for Earth Dams
1.2.14 Filter Design

1.2.11 Plotting of Phreatic Line for Seepage through Earth Dams
For construction of flow nets for seepage through earth dams, the phreatic line needs to be
established first. This is usually done by the method proposed by Casagrande (1937) and is shown in
Figure 2.60a. Note that      in Figure 2.60a is the actual phreatic line. The curve          is a
parabola with its focus at  ; the phreatic line coincides with this parabola, but with some deviations
at the upstream and the downstream force. At a point  , the phreatic line starts at an angle of

to
the upstream face of the dam and

.

Figure 2.60  Determination of phreatic line for seepage through an earth dam
Page 2

Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                    1
Module 2
Lecture 10
Permeability and Seepage -6
Topics
1.2.11 Plotting of Phreatic Line for Seepage through Earth Dams
1.2.12 Entrance, Discharge, and Transfer Conditions of Line of Seepage
through Earth Dams
1.2.13 Flow net Construction for Earth Dams
1.2.14 Filter Design

1.2.11 Plotting of Phreatic Line for Seepage through Earth Dams
For construction of flow nets for seepage through earth dams, the phreatic line needs to be
established first. This is usually done by the method proposed by Casagrande (1937) and is shown in
Figure 2.60a. Note that      in Figure 2.60a is the actual phreatic line. The curve          is a
parabola with its focus at  ; the phreatic line coincides with this parabola, but with some deviations
at the upstream and the downstream force. At a point  , the phreatic line starts at an angle of

to
the upstream face of the dam and

.

Figure 2.60  Determination of phreatic line for seepage through an earth dam

Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                    2

The parabola          can be constructed as follows:
1. Let the distance     be equal to  . Now referring to Figure 2.60b,        based on the
properties of a parabola),

, thus

(2.196)
At          substituting these conditions into equation (2.196) and rearranging, we
obtain

(2.197)
Since         are known, the value of   can be calculated.

2. From equation (2.196),

(2.198)

With   known, the values of   for various values of   can be calculated from equation
(2.198) and the parabola can be constructed.
To complete the phreatic line, the portion    has to be approximated and drawn by hand. When

, the value of   can be calculated from equation (2.183) as

Note that      in Figure 2.60a. Once point   has been located, the curve    can be approximately
drawn by hand.
Figure 2.61 Plot of           against downstream slope angle. (After A. Casagrande,
Seepage through Dams. Contribution to Soil Mechanics, 1925-1940, Boston Society of
Civil Enginering, Boston, 1937.)
Page 3

Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                    1
Module 2
Lecture 10
Permeability and Seepage -6
Topics
1.2.11 Plotting of Phreatic Line for Seepage through Earth Dams
1.2.12 Entrance, Discharge, and Transfer Conditions of Line of Seepage
through Earth Dams
1.2.13 Flow net Construction for Earth Dams
1.2.14 Filter Design

1.2.11 Plotting of Phreatic Line for Seepage through Earth Dams
For construction of flow nets for seepage through earth dams, the phreatic line needs to be
established first. This is usually done by the method proposed by Casagrande (1937) and is shown in
Figure 2.60a. Note that      in Figure 2.60a is the actual phreatic line. The curve          is a
parabola with its focus at  ; the phreatic line coincides with this parabola, but with some deviations
at the upstream and the downstream force. At a point  , the phreatic line starts at an angle of

to
the upstream face of the dam and

.

Figure 2.60  Determination of phreatic line for seepage through an earth dam

Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                    2

The parabola          can be constructed as follows:
1. Let the distance     be equal to  . Now referring to Figure 2.60b,        based on the
properties of a parabola),

, thus

(2.196)
At          substituting these conditions into equation (2.196) and rearranging, we
obtain

(2.197)
Since         are known, the value of   can be calculated.

2. From equation (2.196),

(2.198)

With   known, the values of   for various values of   can be calculated from equation
(2.198) and the parabola can be constructed.
To complete the phreatic line, the portion    has to be approximated and drawn by hand. When

, the value of   can be calculated from equation (2.183) as

Note that      in Figure 2.60a. Once point   has been located, the curve    can be approximately
drawn by hand.
Figure 2.61 Plot of           against downstream slope angle. (After A. Casagrande,
Seepage through Dams. Contribution to Soil Mechanics, 1925-1940, Boston Society of
Civil Enginering, Boston, 1937.)

Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                    3
If

, Casagrande proposed that the value of   can be determined by using the graph given in
Figure 2.61. In Figure 2.60a,

. After locating the point   on the downstream
face, the curve    can be approximately drawn by hand.
Example 1.10. An earth dam section is shown in Figure 2.62. Plot the phreatic line for seepage. For
the earth dam section,

.
Solution

And

From equation (2.197),

1

Using equation (2.198), we can now determine the coordinates of several points of the parabola

70 166
65 142.1
60 120.04
55 99.73
50 81.2
45 64.42
Using the values of   and corresponding   calculated in the above table, the basic parabola has been
plotted in Figure 2.62.
Page 4

Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                    1
Module 2
Lecture 10
Permeability and Seepage -6
Topics
1.2.11 Plotting of Phreatic Line for Seepage through Earth Dams
1.2.12 Entrance, Discharge, and Transfer Conditions of Line of Seepage
through Earth Dams
1.2.13 Flow net Construction for Earth Dams
1.2.14 Filter Design

1.2.11 Plotting of Phreatic Line for Seepage through Earth Dams
For construction of flow nets for seepage through earth dams, the phreatic line needs to be
established first. This is usually done by the method proposed by Casagrande (1937) and is shown in
Figure 2.60a. Note that      in Figure 2.60a is the actual phreatic line. The curve          is a
parabola with its focus at  ; the phreatic line coincides with this parabola, but with some deviations
at the upstream and the downstream force. At a point  , the phreatic line starts at an angle of

to
the upstream face of the dam and

.

Figure 2.60  Determination of phreatic line for seepage through an earth dam

Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                    2

The parabola          can be constructed as follows:
1. Let the distance     be equal to  . Now referring to Figure 2.60b,        based on the
properties of a parabola),

, thus

(2.196)
At          substituting these conditions into equation (2.196) and rearranging, we
obtain

(2.197)
Since         are known, the value of   can be calculated.

2. From equation (2.196),

(2.198)

With   known, the values of   for various values of   can be calculated from equation
(2.198) and the parabola can be constructed.
To complete the phreatic line, the portion    has to be approximated and drawn by hand. When

, the value of   can be calculated from equation (2.183) as

Note that      in Figure 2.60a. Once point   has been located, the curve    can be approximately
drawn by hand.
Figure 2.61 Plot of           against downstream slope angle. (After A. Casagrande,
Seepage through Dams. Contribution to Soil Mechanics, 1925-1940, Boston Society of
Civil Enginering, Boston, 1937.)

Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                    3
If

, Casagrande proposed that the value of   can be determined by using the graph given in
Figure 2.61. In Figure 2.60a,

. After locating the point   on the downstream
face, the curve    can be approximately drawn by hand.
Example 1.10. An earth dam section is shown in Figure 2.62. Plot the phreatic line for seepage. For
the earth dam section,

.
Solution

And

From equation (2.197),

1

Using equation (2.198), we can now determine the coordinates of several points of the parabola

70 166
65 142.1
60 120.04
55 99.73
50 81.2
45 64.42
Using the values of   and corresponding   calculated in the above table, the basic parabola has been
plotted in Figure 2.62.

Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                    4

We calculate   as follows. The equation of the line     can be given by          and the equation
of the parabola [equation (2.198)] is

. The coordinates of point    can be
determined by solving the above two equations:

Or

Hence

The solution of the above equation gives             so

From Figure 2.61, for

,

So,           .
The curve portions           can now be approximately drawn by hand which completes the
phreatic line      Figure 2.62.
Figure 2.62
Page 5

Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                    1
Module 2
Lecture 10
Permeability and Seepage -6
Topics
1.2.11 Plotting of Phreatic Line for Seepage through Earth Dams
1.2.12 Entrance, Discharge, and Transfer Conditions of Line of Seepage
through Earth Dams
1.2.13 Flow net Construction for Earth Dams
1.2.14 Filter Design

1.2.11 Plotting of Phreatic Line for Seepage through Earth Dams
For construction of flow nets for seepage through earth dams, the phreatic line needs to be
established first. This is usually done by the method proposed by Casagrande (1937) and is shown in
Figure 2.60a. Note that      in Figure 2.60a is the actual phreatic line. The curve          is a
parabola with its focus at  ; the phreatic line coincides with this parabola, but with some deviations
at the upstream and the downstream force. At a point  , the phreatic line starts at an angle of

to
the upstream face of the dam and

.

Figure 2.60  Determination of phreatic line for seepage through an earth dam

Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                    2

The parabola          can be constructed as follows:
1. Let the distance     be equal to  . Now referring to Figure 2.60b,        based on the
properties of a parabola),

, thus

(2.196)
At          substituting these conditions into equation (2.196) and rearranging, we
obtain

(2.197)
Since         are known, the value of   can be calculated.

2. From equation (2.196),

(2.198)

With   known, the values of   for various values of   can be calculated from equation
(2.198) and the parabola can be constructed.
To complete the phreatic line, the portion    has to be approximated and drawn by hand. When

, the value of   can be calculated from equation (2.183) as

Note that      in Figure 2.60a. Once point   has been located, the curve    can be approximately
drawn by hand.
Figure 2.61 Plot of           against downstream slope angle. (After A. Casagrande,
Seepage through Dams. Contribution to Soil Mechanics, 1925-1940, Boston Society of
Civil Enginering, Boston, 1937.)

Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                    3
If

, Casagrande proposed that the value of   can be determined by using the graph given in
Figure 2.61. In Figure 2.60a,

. After locating the point   on the downstream
face, the curve    can be approximately drawn by hand.
Example 1.10. An earth dam section is shown in Figure 2.62. Plot the phreatic line for seepage. For
the earth dam section,

.
Solution

And

From equation (2.197),

1

Using equation (2.198), we can now determine the coordinates of several points of the parabola

70 166
65 142.1
60 120.04
55 99.73
50 81.2
45 64.42
Using the values of   and corresponding   calculated in the above table, the basic parabola has been
plotted in Figure 2.62.

Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                    4

We calculate   as follows. The equation of the line     can be given by          and the equation
of the parabola [equation (2.198)] is

. The coordinates of point    can be
determined by solving the above two equations:

Or

Hence

The solution of the above equation gives             so

From Figure 2.61, for

,

So,           .
The curve portions           can now be approximately drawn by hand which completes the
phreatic line      Figure 2.62.
Figure 2.62

Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                    5
1.2.12 Entrance, Discharge, and Transfer Conditions of Line of Seepage
through Earth Dams
A.Casagrande (1937) analyzed the entrance, discharge, and transfer conditions for the line of seepage
through earth dams. When we consider the flow from a free draining material (coefficient of
permeability very large;

into a material of permeability

it is called an entrance.
Similarly, when the flow is from a material of permeability

into a free draining material (

)
it is referred to as discharge. Figure 2.63 shows various entrances, discharge, and transfer
conditions. The transfer conditions show the nature of deflection of the line of seepage when passing
from a materials of permeability

.

Using the conditions given in Figure 2.63 we can determine the nature of the phreatic lines for
various types of earth dam sections.
1.2.13 Flow net Construction for Earth Dams
With knowledge of the nature of the phreatic line and the entrance, discharge, and transfer
conditions, we can proceed to draw flow nets for earth dam sections. Figure 2.64 shows an earth
Figure 2.63 Entrance, discharge, and transfer conditions.. (After A. Casagrande, Seepage
through Dams, Contribution to Soil Mechanics, 1925-1940, Boston.
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