1. GLE/CEE 330: Soil Mechanics
Seepage through Soils
Dante Fratta
Geological Engineering

2. Content Introduction and Objectives Laplace’s Equation
Solutions for the Laplace’s Equation
Flow Nets
Heat and Electrical Flow
Physical Models
Methods of Fragments
Finite Difference Method
Seepage Control: Filters

3. Introduction and Objectives
Type of Problems
Hydrostatic
Seepage
Consolidation
(Atkinson 2007)

4. Introduction and Objectives
Bernoulli’s equation
Flow is governed by total head!!
Swimming pool
Datum
hz1
hp1
hT1
hp2
hz2
hT2
Datum
hz1
hp1
hp2
hz2
hT2
hT1
Open Channel
Filter
hp2
hp2
Datum
hT = hz + hp + hk

5. Introduction and Objectives
To obtain pore pressure (stability analysis)
To calculate flow
To verify piping conditions
Retaining wall
Cofferdam
Hydraulic dam
Sheet pile
Drainage pipe

6. Introduction and Objectives
Type of seepage flow
Confined seepage
Unconfined seepage
(Atkinson 2007)

7. Laplace’s Equation Elemental Cube: Continuity: Saturation S=100 %
Void ratio e=constant
Laminar flow
Continuity: dx dy

8. Laplace’s Equation Continuity: Darcy’s law: Replacing:
if kx=ky Laplace’s Equation!
(isotropic):

9. Laplace’s Equation Typical cases 1 Dimensional: 2-Dimensional:
linear variation!!
2-Dimensional:
3-Dimensional:

10. Laplace’s Equation Solutions
Exact solutions (for simple B.C.’s)
Physical models (scaling problems)
Approximate solutions: method of fragments
Analogies: heat flow and electrical flow
Graphical solutions: flow nets
Numerical solutions: finite differences

11. Flow Nets
The procedure consists on drawing a set of perpendicular lines: equi-potentials and flow lines.
These sets of lines are the solution to the Laplace’s equation.
It is an iterative (and tedious!) process.
Identify boundaries:
First and last equi-potentials
First and last flow lines

12. Flow Nets gradient: flow per channel: total flow: q Flow channel db
Equipotential lines
ds= l db q
hT= equipotential drop
gradient:
flow per channel:
total flow:
Nf = # of flow channels
Ned = # of equipotential drops

13. Flow Nets Example: Confined flow Drawn to Scale! A B Datum Hydraulic
conductivity:
k = 10-6 m/s
h= 9.0 m
11.0 m
2.0 m
16.0 m
2.0 m
exit
gradient A B
10.0 m
Datum
Drawn to Scale!

14. Flow Nets Example Hydraulic conductivity: k = 10-6 m/s
First equipotential
2.0 m
Last equipotential
First flow line
10.0 m
Last flow line

15. Flow Nets Example Hydraulic conductivity: k = 10-6 m/s 11.0 m 2.0 m

16. Flow Nets Example exit gradient A B Hydraulic conductivity
k = 10-6 m/s
11.0 m
2.0 m
16.0 m
exit gradient A B
10.0 m

17. Flow Nets
Example
Seepage loss under the dam:
Exit gradient:

18. Flow Nets Example Total head at points A and B:
Pressure head at points A and B:

19. Flow Nets
Example: Unconfined flow

20. Bibliography
Atkinson, J. (2007). The Mechanics of Soils and Foundations. Taylor & Francis.
Coduto, D. (1999). Geotechnical Engineering. Principles and Practice. Prentice-Hall.
Craig, R. F. (1987). Soil Mechanics. Chapman & Hall.
Encyclopædia Britannica (2000).
Holtz, R. and Kovaks, D. (1981). An Introduction to Geotechnical Engineering. Prentice-Hall.
McCarthy, D. (1998). Essential of Soil Mechanics and Foundation. Prentice-Hall.