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

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

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

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

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

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

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

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

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

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

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

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

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!

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

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

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

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

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

Flow Nets Example: Unconfined flow

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). http://www.encyclopedia.com. Holtz, R. and Kovaks, D. (1981). An Introduction to Geotechnical Engineering. Prentice-Hall. McCarthy, D. (1998). Essential of Soil Mechanics and Foundation. Prentice-Hall.