Soil Mechanicsustce424soilmechanics01.wikispaces.com/file/view/11_Permeability... · Seepage Terminology Seepage is defined as the flow of a fluid, usually water, through a soil under

XI. PERMEABILITY Soil Mechanics

SEEPAGE AND FLOW NETS

Seepage Terminology Seepage is defined as the flow of a fluid, usually water,

through a soil under a hydraulic gradient.

A hydraulic gradient (i) is supposed to exist between two

points if there exists a difference in the hydraulic head at

the two points.

Hydraulic head is the sum of the position or datum head

and pressure head of water. The discussion on flow nets and seepage relates to the practical aspect of

controlling groundwater during and after construction of foundation below the groundwater table, earth dam

and weirs on permeable foundations.

XI.1. PERMEABILITY Soil Mechanics

SEEPAGE AND FLOW NETS One-dimensional flow

Flow Net for One-Dimensional Flow

Figure 6.1(a) shows a tube of square cross-section (400 mm × 400 mm)

through which steady state vertical flow is occurring. The total head, elevation head and pressure are plotted in Fig. 6.1(b). The rate of

seepage through the tube may be computed by Darcy’s law:

as the situation is one of simple one-dimensional flow.



In the figure, dashed lines indicate the lines along which the total head is a

constant. These lines through points of equal total head are known as

‘equipotential lines’. Just as the number of flow lines is infinite, the number of

equipotential lines is also infinite.

If equipotential lines are drawn at equal intervals, it means that the head loss

between any two consecutive equipotential lines is the same.

A system of flow lines and equipotential lines, as shown in Fig. 6.1 (c),

constitutes a ‘flow net’ . In isotropic soil, the flow lines and equipotential lines

intersect at right angles, indicating that the direction of flow is perpendicular

to the equipotential lines. An orthogonal net is formed by the intersecting

flow lines and equipotential lines. The simplest of such patterns is one of the

squares. From a flow net three very useful items of information may be

obtained: rate of flow or discharge; head; and hydraulic gradient.


First, let us see how to determine the rate of flow or discharge from the flow net. Consider square a

in the flow net – Fig. 6.1(c). The discharge 𝑞𝑎 through this square is,

𝑞𝑎 = 𝑘. 𝑖𝑎. 𝐴𝑎

The head lost in square a is given as 𝐻/𝑁𝑑, where 𝐻 is the total head lost and 𝑁𝑑 is the number of

head drops in the flow net. 𝑖𝑎 is then equal to 𝐻/(𝑁𝑑 . 𝑙), where 𝑙 is the vertical dimension of square 𝑎. The cross-sectional area 𝐴𝑎 of square 𝑎, as seen in plan, is 𝑏 as shown in the figure, since a unit

dimension perpendicular to the plane of the paper is to be considered for the sake of convenience. Thus,

𝑞𝑎 = 𝑘.𝐻

𝑁𝑑 . 𝑙. 𝑏

Since a square net is chosen, 𝑏 = 𝑙. Thus,

𝑞𝑎 = 𝑘.𝐻

𝑁𝑑

If the number of flow channels in a flow net is equal to 𝑁𝑓, the total rate of flow through all the

channels per unit length can be given by,

𝒒 = 𝒌.𝑯.𝑵𝒇

𝑵𝒅


SEEPAGE AND FLOW NETS Two-dimensional flow

Seepage Terminology

concrete dam

impervious strata

soil

Stream line is simply the path of a water molecule.

datum

H

h = 0 h = H

From upstream to downstream, total head steadily decreases along the stream line.

Equipotential line is simply a contour of constant total head.

concrete dam

impervious strata

soil

datum

H

h = 0 h = H

h=0.8H

Seepage Terminology

Flow Net A network of selected stream lines and equipotential lines.

concrete dam

impervious strata

soil

curvilinear

square

90º

Quantity of Seepage (q)

d

f

N

NHkq .. …. per unit length normal to the plane

# of flow channels

# of equipotential drops

impervious strata

concrete dam

H

head loss from upstream to

downstream

Heads at a Point ‘X’

impervious strata

concrete dam datum

X

z

H

h = H h = 0

Total head, h = H - # of drops from upstream x h

h

Elevation head, he = - z

Pressure head, hp = Total head – Elevation head dN

H

Flow Net for Two-Dimensional Flow

Flow under Sheet Pile Wall


Flow under Concrete Dam


Flow under Concrete Dam


Flow through Earth Dam





Basic Equation for Seepage

Laplace’s Equation of Continuity

The flow net was introduced in an intuitive manner in the preceding sections. The

equation for seepage through soil which forms the theoretical basis for the flow net as

well as other methods of solving flow problems will be derived in this section.

The following assumptions are made:

1. Darcy’s law is valid for flow through soil.

2. The hydraulic boundary conditions are known at entry and exit of the fluid (water) into the porous

medium (soil).

3. Water is incompressible.

4. The porous medium is incompressible.

These assumptions have been known to be very nearly or precisely valid.





For flow at a point A, we consider an elemental soil block. The block has dimensions dx,

dy, and dz (length dy is perpendicular to the plane of the paper). Let vx and vz be the

components of the discharge velocity in the horizontal and vertical directions,

respectively. The rate of flow of water into the elemental block in the horizontal direction is equal to vx.dz.dy, and in the vertical direction it is vz.dx.dy. The rates of outflow

from the block in the horizontal and vertical directions are:

Assuming that water is incompressible and that no volume change in the soil mass

occurs, we know that the total rate of inflow should equal the total rate of outflow.

Thus,



With Darcy’s law, the discharge velocities can be expressed as:

where 𝑘𝑥 and 𝑘𝑧 are the hydraulic conductivities in the vertical and horizontal

directions, respectively.

If the soil is isotropic with respect to the hydraulic conductivity – that is, 𝑘𝑥 = 𝑘𝑧 - the

continuity equation for two-dimensional flow simplifies to.

𝒒 = 𝒌.𝑯.𝑵𝒇

𝑵𝒅



With Darcy’s law, the discharge velocities can be expressed as:

where 𝑘𝑥 and 𝑘𝑧 are the hydraulic conductivities in the vertical and horizontal

directions, respectively.

For anisotropic soils, 𝑘𝑥 ≠ 𝑘𝑧. In this case, the equation represents two families of curves

that do not meet at 90°. However, we can rewrite the equation as:

𝒒 = 𝒌𝒙. 𝒌𝒛. 𝑯.𝑵𝒇

𝑵𝒅


SEEPAGE AND FLOW NETS Safety of Hydraulic Structures

Against Piping

Piping in Granular Soils

datum concrete dam

impervious strata

soil

H

At the downstream, near the dam,

h = the head loss between the last

two equipotential lines

l

l

hiexit

the exit hydraulic gradient,

l = the length of the flow element

Piping in Granular Soils

datum

concrete dam

impervious strata

soil

H

If iexit exceeds the critical hydraulic gradient (icr), firstly, the

soil grains at exit get washed away.

no soil; all water

This phenomenon progresses, forming a free passage of water (pipe). If effective stress equal zero, soil stability is

lost leading to ‘boiling’ or ‘quick condition’.

w

cri

'

Piping in Granular Soils Piping is a very serious problem. It leads to downstream

flooding which can result in loss of lives.

concrete dam

impervious strata

soil

Therefore, provide adequate safety factor against piping.

exit

crpiping

i

iSF ..

typically 3~4

Stresses due to Flow

X

soil

hw

L

Static Situation (No flow)

z v = whw + satz

u = w (hw + z)

v ' = ' z

At X,

Introduction Introduction

Downward Flow

hw

L

flow

X

soil

z

v = whw + satz

w hw + w(L-H)(z/L)

v ' = ' z + wiz

At X,

H u = w hw

u = w (hw+L-H)

… as for static case

= w hw + w(z-iz)

= w (hw+z) - wiz Reduction due to flow

Increase due to flow

u =

Stresses due to Flow Introduction Introduction

flow Upward Flow

hw

L X

soil

z

v = whw + satz

= w hw + w(L+H)(z/L)

v ' = ' z - wiz

At X, H

u = w hw

u = w (hw+L+H)

… as for static case

= w hw + w(z+iz)

= w (hw+z) + wiz

Increase due to flow

Reduction due to flow

u

Stresses due to Flow Introduction Introduction

Quick Condition in Granular Soils During upward flow, at X:

v ' = ' z - wiz

flow

hw

L X

soil

z

H

izw

w

'

Critical hydraulic gradient (icr)

If i > icr, the effective stress is negative.

i.e., no intergranular contact, thus, failure.

- Quick Condition -

Problem 1 Draw the total, elevation and pressure heads

diagrams for the figure shown. Also,

determine the following:

1.1 Hydraulic gradient (2 m/m)

1.2 Discharge velocity (1 cm/sec)

1.3 Seepage velocity (3 cm/sec)

Problem Set 8

1.2 m

1.8 m

0.6 m

Problem 2 Draw the total, elevation and pressure heads

diagrams for the figure shown.

Problem Set 8

1.2 m

1.8 m

0.6 m

1.2 m

Problem 3 A flow net for flow around a single row

of sheet piles in a permeable soil

layer is shown in the figure. We are given that kx = kz = k = 5x10-3 cm/sec.

Determine:

3.1 How high (above the ground surface) will

the water rise if piezometers are placed at

points a, b, c, and d?

3.2 What is the rate of seepage through flow

channel II per unit length (perpendicular to the

section shown)?

Problem Set 8

Problem 4 A flow net for flow around a

single row of sheet piles in a

permeable soil layer is shown

in the figure. We are given that kx = kz = k = 5x10-9 m/sec.

Determine:

4.1 The pressure at a ~ l.

4.2 The total flow rate.

4.3 The hydraulic gradient at ‘r’

element.

Problem Set 8

El. 0 m

El. 11.7 m

El. 14.7 m

El. 18 m

El. 27 m

El. 19.5 m

El. 9 m

c

d f

g

Problem 5 A flow net for flow around a

concrete in a permeable

soil layer is shown in the figure. Determine the uplift

pressure under the concrete

dam.

Problem Set 8

impervious strata

concrete dam

14 m

El. 0 m

El. 10 m

El. 17 m

El. 8 m a b c d e f

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Soil Mechanicsustce424soilmechanics01.wikispaces.com/file/view/11_Permeability... · Seepage Terminology Seepage is defined as the flow of a fluid, usually water, through a soil under