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


NPTEL- Advanced Geotechnical Engineering 
 
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


NPTEL- Advanced Geotechnical Engineering 
 
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 
NPTEL- Advanced Geotechnical Engineering 
 
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


NPTEL- Advanced Geotechnical Engineering 
 
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 
NPTEL- Advanced Geotechnical Engineering 
 
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.) 
NPTEL- Advanced Geotechnical Engineering 
 
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


NPTEL- Advanced Geotechnical Engineering 
 
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 
NPTEL- Advanced Geotechnical Engineering 
 
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.) 
NPTEL- Advanced Geotechnical Engineering 
 
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. 
NPTEL- Advanced Geotechnical Engineering 
 
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


NPTEL- Advanced Geotechnical Engineering 
 
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 
NPTEL- Advanced Geotechnical Engineering 
 
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.) 
NPTEL- Advanced Geotechnical Engineering 
 
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. 
NPTEL- Advanced Geotechnical Engineering 
 
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 
NPTEL- Advanced Geotechnical Engineering 
 
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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