## Permeability and Seepage

### 1. Introduction

• Water flows through the voids in a soil which are interconnected. This flow may be called seepage, since the velocities are very small.
• Water flows from a higher energy to a lower energy and behaves according to the principles of fluid mechanics.

### 2. Energy Considerations from Fluid Mechanics

#### B. An Energy Equation for Water Flow in Soils

The velocities of water flowing through the voids in a soil are very small, and the velocity head in the previous equation may be neglected. Therefore, for flow of water in soil the equation is:

#### C. Energy Loss Equation

Water is flowing in the direction indicated in the figure.

### 3. Darcy's Law and Other Principles

#### C. Seepage Velocity

Only the portion of the soil cross-sectional area representing the void spaces has water flowing. The solid portion of the area will not allow flow. Flow rate calculations are done using the average velocity, v, and the total cross-sectional area, A. If we let seepage velocity represent the actual velocity of water flowing through the void spaces, then the following equation may be developed.

### 5. Coefficient of Permeability

The notation for coefficient of permeability is k. It is sometimes called hydraulic conductivity.

#### A. Typical Values

 k k Soil Type cm/sec ft/min Clean Gravel 1.0 to 100 2.0 to 200 Coarse Sand 0.01 to 1.0 0.02 to 2.0 Fine Sand 0.001 to 0.01 0.002 to 0.02 Silty Clay 0.00001 to 0.001 0.00002 to 0.002 Clay Less Than 0.000001 Less Than 0.000002

#### B. Laboratory Tests to Determine Coefficient of Permeabilty

• Constant Head Permeability Test.
• Falling Head Permeability Test.
• Both of these tests will be conducted as laboratory exercises.
• Descriptions of the tests are in the textbook.

#### C. Empirical Relations for Coefficient of Permeability

Several empirical equations have been proposed. Probably the most well known is Hazen's approximation.

### 6. Equivalent Permeability in Stratified Soil

In nature, soil is normally stratified or layered. The layers are usually horizontal as most soils are deposited in this manner.

#### A. Flow Parallel to the Layers

The figure shows horizontal layers of soil with flow parallel to the layers. Following the figure, an equation is developed to calculate the equivalent permeability for flow parallel to the layers.

#### B. Flow Perpendicular to the Layers

The figure shows horizontal layers of soil with flow perpendicular to the layers. Following the figure, an equation is developed to calculate the equivalent permeability for flow perpendicular to the layers.

### 8. Calculation of Heads

#### A. Basic Principles

• Total Head = Pressure Head + Elevation Head
• The pressure head is zero at a water surface.
• The head loss in the water is assumed to be zero.
• All head loss occurs in the soil.

### 9.  Laplace's Equation of Continuity

#### C.  Solutions to Laplace's Equation

• Computer solutions using finite element or finite difference techniques.
• Graphical solutions known as flow nets.

### 10.  Flow Net Construction

#### A.  An Example

This example for constructing a flow net is similar to Figure 5.21 in the textbook.  In the example, water is flowing under a sheet pile wall that is driven partially through a permeable isotropic soil.  The first step is to draw the situation to scale.

#### B.  Flow Lines and Equipotential Lines

• A flow line is the path a water particle would follow in moving from upstream to downstream.
• An equipotential line is a line along which the total head, h, is a constant value.  It is similar to a contour line, except that total head is constant, rather than elevation.

#### C.  A Flow Net

A flow net is a combination of flow lines and equipotential lines that satisfy Laplace's equation and the boundary conditions.

#### D.  Rules for Flow Net Construction

1. Draw the situation to scale.
2. Establish the equipotential boundary conditions.  For this example, ba and de are boundary equipotential lines.  The total head is 56' everywhere along line ba.  The total head is 39' everywhere along line de.
3. Establish the flow boundary conditions.  For this example, acd and fg are boundary flow lines.
4. Flow lines and equipotential lines must always intersect at right angles.
5. The fields should be "square".  By this we mean that the length and width of a field should be equal.  Fields are the spaces in a flow net formed by the intersecting flow lines and equipotential lines.  The sides of the fields are commonly curved, so the term "square" is a stretch of the normal definition.

#### E.  Construction of the Flow Net

The flow net for this example follows.  Equipotential lines are red and dashed, while flow lines are green and solid.  The length and width of one of the fields is shown.  The length in the direction of flow is L, while the width is b.  To be "square" ( L= b ).

### 11.  Flow Net Calculations

#### A.  Example

The flow net from the previous section will be used for the calculations.  The coefficient of permeability for the soil is   0.04 ft/s

B.  Flow Rate

#### C.  Heads and Water Pressures

We will calculate the water pressure at point A in the flow net.

#### D.  Gradients

Gradients are different for each field in a flow net.  Head losses across each field will be the same, but the flow length depends on the size of the field.  Small fields will have a small flow length and a large gradient.  Conversely, large fields will have a large flow length and small gradients.  Flow length may be measured directly from the flow net, since it is drawn to scale.  Following is a gradient calculation for the field with " L = 11' " printed inside.  The length, L, was determined by direct measurement and multiplication by the scale factor.