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1. INTRODUCTION

As water can flow easily through interconnected pores in soil media, soil is

permeable. Flow occurs from high energy point to low energy point. Depending on

the pore size the flow may be steady or unsteady, laminar or turbulent. For simplicity

the flow is generally considered as laminar and steady flow. The study of flow of

water through soil is necessary for estimating the quantity of underground seepage

under various hydraulic conditions, for investigating problems involving the pumping

of water for underground construction, for obtaining the amount of seepage or

leakage below the dam, the uplift pressure caused by the water on the base of the

concrete dam, and the possibility of danger of a quick or liquefaction condition at

points where seepage water comes to the ground surface.

1.1 Bernoulli’s Equation

According to Bernoulli’s Equation the total head (energy per unit weight) at a point in

water under motion can be given by the following equation-

where = total head ; = pressure ; v = velocity ; = gravitational acceleration ; =

unit weight of water ; = elevation of a point above or below the datum level.

Generally seepage velocity is very

small such that its contribution to the

total head is very less or negligible.

Hence total head may be presented

as –

Now as per the fig the head loss

between A and B may be written as-

1.2 Darcy’s Law

The pores of most soils are so small that flow of water through them is laminar.

However in coarse soils flow of water may be turbulent. The pore channels of a soil

mass are so narrow and tortuous, so irregular in cross section that exact analysis of

the flow through individual pores is very cumbersome. But flow through the individual

pores is not our interest. Instead we are interested in combined or overall flow

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through all pores of an

element of volume which is

sufficiently large to give a

typical representation of the

soil mass as a whole.

Consider an element with

cross section and length

, the element is inclined at an angle to the horizontal; inflows and outflows only

occurs at two ends. The following forces are acting on the element-

1. Force on the left hand face and the right hand face due to water pressure.

2. Gravitational force equal to times the volume of water.

3. Frictional force due to viscosity.

Resolving the forces in the direction of L,

Where, is effective porosity.Substituting

and rearranging

Darcy’s experiments showed that for low velocities, the resisting force is proportional

to the velocity ; for mathematical convenience this is written as

, where

is dynamic viscosity, is the intrinsic permeability of the porous material. Hence,

, where k is the hydraulic conductivity defined as

.

This equation is known as Darcy’s law. In soil mechanics the word permeability is

often used as an alternative of hydraulic conductivity.

Now discharge,

. Here in the previous equation is total cross-sectional

area of the soil mass,across which the discharge occurs and is the superficial

velocity or apparent velocity. Hence careful distinction between the superficial

velocity and the seepage velocity vs is important. The discharge may be written as

, where area of void , porosity.

The range of validity of Darcy’s Law has been studied by several investigators.

Arough but satisfactory criterion of the limit of applicability of Darcy’s Law, given by

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Fancher, Lewis, and Barnes is

i.e. Reynolds number 1

Where, = superficial velocity ; diameter of the sphere which has a volume equal

to the quotient of the volume of solids of a sample and the number of grains in the

sample.

1.3 Stream function & velocity potential

An element of fluid (control volume) is shown in the above figure. At the centre of the

element the density is , and the velocity has components , and . Now we are

considering the flow in each direction separately. represents the x-component of

mass rate of flow per unit area at the centre of the element. Hence on the right face

and on the left face

Here we are neglecting the higher order terms of Taylor series expansion. Now net

rate of mass outflow in the x direction

Similarly net ret mass outflow in y direction

andin z direction

Now change of mass of the control volume due to change in the density=

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Now from the above discussion we can conclude that the differential equation of

conservation of mass is

Here we concern with steady flow (i.e.

) and incompressible flow (i.e. =

constant). Hence the above equation simplifies to the following equation

The above equation is applicable for both steady and unsteady flow of

incompressible fluid. Now for two dimensional flow the equation becomes

Now we have two variables, u and v, to deal with. Now if we define a function ψ(x,y),

called the stream function which relates the two components of velocity as shown

below

and

then the continuity equation is identically satisfied. In this way we have simplified the

analysis by having to determine only one unknown function, , rather than the

two functions, and .

If we move from a point to neighborhood point , the change in

the value of is as follows

Along a line of constant we have , so that v.dx + u.dy = 0 and, therefore,

along a line of constant ,

,which is the defining equation for a streamline.

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Thus we can conclude that lines along which is

constant are streamlines. Hence if we know the

function , we can plot lines of constant to

provide the family of streamlines that are helpful in

visualizing the pattern of flow.

Consider two closely spaced streamlines, shown in the

figure of previous page. Let represents the volume

rate of flow (per unit width perpendicular to the x-y

plane) passing between the two stream lines. From

conservation of mass we can conclude that

Thus volume rate of flow, q, between two streamlines such as , can be

determined as

Hence the relative value of with respect to determines the direction of flow as

shown in the figure at right to the page.

Now many flows have negligible or zero vorticity and are called irrotational flow.

For threedimensional irrotational flow curlV = 0.

Now if we define a scalar function such that

so

that the above equations are identically satisfied. Here φ is called the velocity

potential which is applicable for irrotational flow only.

Now along a constant φ line for two dimensional flow

. Hence

From the above equations we can conclude that equipotential lines (φ lines) are

orthogonal to the stream lines( lines). For any potential flow field a “flow net ” can

be drawn that consists of a family of streamlines and equipotential lines.

1.4 Khosla’s Theory

Bligh’s theory states that water creeps along the bottom contour of the structure

which fails to describe actual phenomenon of seepage. Khosla’s theory, a more

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appropriate theory, can be

summarized as follows

a) Water seeps

through subsoil

along a set of

stream lines. This

steady seepage in

a vertical plane for

an isotropic soil

can be expressed

by Laplacian

equation which will

be discussed in

section 2.1.3.the first streamline follows the bottom contour of the structure as

stated in Bligh’s theory. The remaining streamlines follows smooth transition

curves as shown in the above figure by deep firm line. The dotted lines are

equipotential lines.

b) The seepage water exerts force at each point in the direction of flow and

tangential to the flow. This force has an upward component where the

streamlines turn upward. For the soil grains to remain stable the upward force

must be nullified by the submerged weight of soil grains. Since the direction of

streamline at the exit is vertical, the force has maximum disturbing tendency

at the exit. The disturbing force at any point is proportional to the gradient of

total head or rate of head loss. The gradient of total head of water at the exit

is known as exit gradient. The exit gradient should be below a typical limit

such that the upward force will be nullified by the submerged weight of soil.

2. FUNDAMENTALS OF SEEPAGE

In general flow through an earth mass is a three dimensional phenomenon. To

simplify the analysis it can best be shown by using the simple two dimensional case.

Two assumptions made in seepage analysis are that Darcy’s law is valid and that the

soil is homogeneous.

2.1 Concept of flow-net

Flow net is a graphical representation of flow through soil. The path which a water

particle follows during seepage through a saturated soil mass is called flow line or

stream line which has been already discussed in the earlier section. Now we will

discuss various aspects of flow nets.

2.1.1 Boundary conditions

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If fixed conditions is retained at all points of the boundary of a typical cross sections

of a soil mass, the flow net is uniquely determined, i.e. one and only one solution can

exists. If the boundary conditions were to be slightly disturbed or changed, a certain

time would be required for flow to readjust itself to a steady state. A long interval of

time may be elapsed before a steady state is attained if the soil is highly impervious

and very compressible. The expression of boundary conditions consists of statement

of head or flow conditions at all boundary points. The following figure illustrate this

concept:

The boundary conditions of the above figure are completely defined by four following

statements:

1. Line AB is an equi-potential line along which head equals ht.

2. Line CD is an equi-potential line along which head equals zero if we take CD

as datum level.

3. From point A, which touches the piling at the surface of the soil, a line

following the impervious surface of the pile down one side to E and then up on

the other side to C is a flow line.

4. The line FG, including its extension to both right and left is a flow line.

2.1.2 Square figures

Previous figure shows that the figures formed by adjacent pairs of flow and

equipotential lines, figure J1K1M1N1 for example, and resemble squares. All four

corners of each such figure form right angles and the mean distances between

opposite faces are alike.Although this does not constitute a square according to the

strict meaning of the word, it may be noted that if such a figure is subdivided into four

figures and then is further subdivided, there is an approach toward true squares.

Thus the use of a very large number of closely spaced lines would truly give

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squares, and in flow net analysis the term square figures has come into common use

for figures of the sizes commonly used.The significance behind the use of square

figures is very important, and it will now be demonstrated.

For two-dimensional flow Darcy’s law may be written as

Where, y is the dimension of the soil mass in the direction normal to the section.

Designating

by q and by b, and defining q as the discharge per running

meter (i.e. per meter normal to the section) and b as the trace of the area gives the

form

This equation may be used to express the discharge through any figure of a flow net.

In the section 2.1.1 the figures J1K1M1N1 and J2K2M2N2 are arbitrarily chosen as any

two figures of the net, and J’K’M’N’, which is bounded by same flow lines as the first

figure and by the same equipotential lines as the second figure, is used as an

auxiliary figure. Darcy’s law for the discharge through any figure may be written

Where represents the head loss in crossing the figure, and dimensions b and

are illustrated in the figure of section 2.1.1. The expressions in this form for each

three figures under consideration are

The value of k is the same for all figures and, since the three figures which are being

analyzed are squares, the ratios,

,

and

are all equal to unity.Since the auxiliary

square has thesame flow line boundaries as the first square, . Since it has

the same equipotential boundaries as the second square, . Hence from

the previous three equations we can write that and .

Thus it is seen that when all figures are squares there must be the same quantity of

flow through each figure and the same head drop in crossing each figure. It follows

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that flows between adjacent pairs of flow lines are alike throughout and potential

differences between successive equipotential lines are alike throughout.

2.1.3 The Laplace Equations

From Darcy’s law we can write

and

.From continuity

equation in two dimensional flow we can write

. No combining these

three equation we can write

Now for isotropic soil, , which results the following conclusion

For square figure, , which results the following conclusion

These above to equations are the Laplacians of conjugate variables, and

.These equations present the fundamental relationships for steady flow based on

Darcy’s Law in isotropic soils, and states simply that the sum of the components of

space rates of gradient change must be zero. In other words, gradient changes in

the x direction must be balanced by gradient changes of opposite sign in the z

direction if the volume is to remain constant.

2.1.4 Scale transformation in account to stratification

However in many practical situations stratification exists. The demonstration which

follows shows that the use of a simple scale transformation leads to a distorted plot

on which the flow net principles can be applied in case of stratification. In the

previous section we have derived the following relationship

This equation may be written as

If is defined as a new coordinate variable, measured in the same direction as x

and equal to x multiplied by a constant, as expressed by

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then it can be written as

This equation is a Laplacian in the coordinate variables and . Thus, if a cross

section through stratified soil is drawn to distorted scale, the true z scale and the

transformed scale being used for coordinates, the flow net may be sketched on it.

It should be noted that, in the above treatment, the direction is defined as the

direction of maximum permeability. In the majority of cases this direction will be

horizontal, but when the maximum permeability is in a diagonal direction the axis

and the transformed or axis must be taken in this diagonal direction.

2.2 Determination of Quantity of Seepage

The quantity of seepage may easily be determined once the flow net is available.

The expression of discharge through any square is

The number of flow paths and the number of equipotential dropsalong each path

may be counted on any flow net; these numbers will be designated, respectively, by

and .Since all paths have same amount of flow and all potential drops are equal,

total seepage, and total head dissipated, .The substitution of

these two expressions into that above leads to

The seepage for any given cross section equals the product of three quantities - a

constant which depends only on the geometry of the crosssection, the effective

permeability of the soil, and the head dissipated in passing through the section.

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Now if the section is

transformed as discussed in

the section 2.1.4, when the

above equation is used for a

transformed section, the

value that applies for

permeability may be

explained by the following

simple demonstration.

Let be the effective permeability for the transformed section. Now flow through the

transformed square shown in the figure is

.

But in case of natural scale flow through the second figure is

.Now

equating the two expressions we can conclude that effective permeability for the

transformed figure, .

Hence the rate of seepage per unit width can be calculated from the following

equation

3. SEEPAGETHROUGH DAM

The main purpose of dam is to reserve water for irrigation, water supply,

hydroelectric power generation, flood control etc. An initial classification of dams

based on the construction material is shown as follows-

Embankment dams are older and simpler in structural concept than early masonry

dams. It utilizes locally available materials. But concrete dams are more demanding

in relation to the foundation condition though advanced and expensive construction

Dams

Embankment dams

Earth fill Rock fill

Concrete dams (including masonry dams)

Gravity Arch Buttress Multiple arch

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skills are required.The effect of seepage is to be considered properly to avoid poor

design. Generally the boundary conditions related to the seepage through the

pervious soil strata beneath the dam structure. But in case of earth fill dam seepage

also occurs through the dam and the uppermost flow line is not readily found out,

thus introducing complication. This upper flow line is free water surface which is

known as line of seepage.

3.1 General considerations of earthen dam

For simplicity we are considering a homogeneous earthen dam as shown in the

above figure. In this case the top flow line or the line of seepage is the flow line

connecting the point B and rock toe. We will discuss how to find out this top flow

line in the next section. The top flow line must obey the following three conditions-

a) As the top flow line is at atmospheric pressure the only type of head can exist

along it is elevation head. Hence, from the properties of square figures as

discussed in section 2.1.2 we can say that there must be constant elevation

drop between the points at which successive equipotentials meet the top flow

line as shown below.

b) Upstream slope( AB) is an equipotential line. So the top flow line must be

normal to the upstream slope at the starting point as shown below.In the second

figure the coarse material at the upstream is so pervious that it does not offer

appreciable resistance to flow. Hence the upstreamequipotential line is the

downstream boundary of coarse material. The top flow

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line cannot be normal to the equipotential line as it cannot rise.

c) In the downstream end the top flow line tends to follow the direction of gravity as

shown below.

The second figure shows a vertical exit condition into a rockfill toe.

3.2 Solution of horizontal under-drainage or Parabolic case

The constituents of flow net i.e. flow lines and equipotential lines are curves resulted

from two conjugate functions - and

. These conjugate functions can be

represented by nests of confocal parabolas. The following figure (a) shows the

general properties of any parabola.

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In figure (b) all curves are parabolas with the common focus F. A cross section for

which such a flow net holds rigorously is given in figure (c).

Here BC and DF are flow lines, and BD and FC are equipotential lines. The

upstream equipotential line is unusual for a homogeneous dam with underdrainage

filter.

The commonest case of upstream equipotential boundary is shown in figure (d). The

flow net for this section will not consist of true parabolas, but there will be close

similarity with them in the downstream portion. The top flow line is close to parabola

except for a short distance of reversed curvature near point B. If we extend this

parabola to the upstream water surface it will intersect the surface at point A.Arthur

Casagrande found that AB equals approximately times BE for dams with

reasonably flat upstream slopes. Here F is the common focus for all parabolas. The

short section of reversed curvature starting at point B may be sketched by eye. Let

focal distance FA be . From figure (a) for any point on the parabola D having co-

ordinate , , which implies the following relation

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Now for point A of figure (d), and , which leads to the following

relationship

The distance can be obtained by graphical construction. From the properties of

parabola we can say that FC = and FG = KJ = . Now when the points C,G,A,B

are located the top flow line can be easily traced out. As FG = , head at G is equal to

. Now the flow net of the region GCFH has and total head, . From

the expression of quantity of total seepage discussed in the section 2.2 we can say

that

3.3 Correction to parabolic case for sloping discharge face

Here the slope of discharge face is designated as α, measured clockwise from

horizontal. Thus the underdrainage case is defined by an α of 180 degrees. The

sketches of flow nets for the downstream portions of dams with α values of

and are shown below.

Here the intersection of bottom flow line with the discharge face has been used as

focus to give the parabolas shown by the dotted lines in these above figures. In each

sketch the top flow line is in close agreement with the parabola except for a short

final portion of its path. Here is the distance between the parabola and the actual

breakout point measured along the discharge face. The distance between actual

breakout point and focus is designated as . The values of /( ) are noted

oneach figure. Figure (d) is based on these values and other values from flow nets

not shown. In the figure (d) the value of /( ) for any slope of or more.

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Now to determine the top flow line graphically for homogeneous dam resting on an

impervious foundation the following steps are to be followed-

a) The section should be drawn in with proper scale factor.

b) Sketch the top flow line parabola with its focus at the breakout point of the

bottom flow line as discussed in the section 3.2.

c) For dams with flat slopes, this parabola will be correct for the central portion of

the top flow line. The portion of the top flow line at entry is sketched visually to

meet the boundary condition there i.e., the perpendicularity with the upstream

face,which is a boundary equipotential and the tangentiality with the base

parabola.

d) The ratio /( ) can be found out from the graph by knowing the slope

of discharge face.

e) The necessary correction at the downstream end may be made by knowing

the values of and ( ).

3.4 The L. Cassagrande solution for a triangular dam

For triangular dams on impervious foundations with discharge faces at 90° or less to

the horizontal L. Casagrandegives a simple and reasonably accurate solution for the

top flow line as shown in the following discussions.

In the following figure the top flow line actually starts at B. Applying the entrance

correction as discussed in section 3.2 we have found the point A as an

assumedinitial point. We have to find the distance of break out point C of top flow

line which is designated as . Here is the vertical distance from the foundation level

and is thedistance measured along the top flow line. As along the top flow line the

only head isthe elevation head, gradient at any point is

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The followings are the two assumptions made in this method –

a) The distance GH is equal to its vertical projection i.e. the coordinate of point

G.

b) Gradient at point G of top flow line i.e. is the average gradient for

all points of equipotential line GH.

No flow per unit width across the equipotential line GH is

At point C the gradient equal , and equals . Thus flow through

equipotential line CF becomes

Now equating above two equations we get

Now at point A, is equal to and is assumed to be zero. The distance along top

flow line from A to D is designated as in the figure. Thus at point C is equal to

and is equal to Now integrating the above equation from A to C we

can get

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By substituting the approximate value of i.e. we can get the final equation

Once the value of has been determined the discharge, q can be easily determined

from the equation This theory can give good result for varies from

to For angle the error occurs due to the assumptions becomes

large and this method is not normally used.

3.5 Schaffernak and Iterson’s solution

If downstream slope, the distance can be found out using Schaffernak and

Iterson’s method as shown below.

Shcaffernak and Iterson (1917) assumed the energy gradient as or

Now flow per unit width through vertical section EJ is given by

But and .

So,

Now integrating between the limits to and to

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Once the value of has been determined the discharge, q can be easily determined

from the equation

3.6 Flow net patterns for concrete dams

Some flow net patterns are shown below for a concrete dam resting on an isotropic

soil with or without cut-off wall. The first one is with no cut-off walls, the second with

cut-off wall at the heel as well as at the toe,the third with cut off-wall at the heel only,

the fourth with cut-off wall at the toe only and thefifth is with upstream impervious

blanket. It is to be noted that the effect of cut-off walls is to reduce the under

seepage, the uplift pressure on the underside of thedam and also the hydraulic

gradient at the exit, called the exit gradient as discussed in section 1.4.

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4. FILTER REQUIREMENTS

Due to seepage the fine particles are washed out i.e. erosion occurs and the pores

of coarse grains are clogged due to which seepage force is increased. Hence coarse

material of proper size is required which is called filter material such that these

problems can be eliminated. So the filter material should fulfill the following

conditions- (a) the size of the voids in the filter material should be small enough to

hold the larger particles of the protected material in place, and (b) the filter material

should have a high hydraulic conductivity to prevent large seepage forces and

hydraulic pressure in the filters.

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Terzaghi and Peck (1948) provided the following to satisfy the first condition

And to satisfy the second condition i.e. condition (b) they gave the following condition

Where diameter through which 15% of filter material will pass;

diameter through which 15% of soil to be protected will pass;

diameter through which 85% of soil to be protected will pass.

The figure shown below simplifies the use of these two equations. Let the material

used in construction of earth dam has the grain size distribution curve . From the

curve we can determine and and plot them as shown in the figure.

The acceptable grain-size distribution of the filter material will have to lie in the

shaded zone.

5. DRAINAGE GALLERY

Seepage exists within every dam. We can only control and direct it. In embankment

dams it is controlled by suitably located pervious zones as discussed in the section

4. In case of concrete dam vertical drains are formed inside the upstream face, and

seepage pressure is relieved into an internal gallery or outlet drain. In the case of

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arch dams, seepage pressures in the rock

abutments are frequently relieved by

systems of bored drains and tunnels.

Water seeping through the dam is directed

from several small collectors into main

drainage gallery. The gallery also serves to

give access for inspection purposes, to

monitor the behavior of dam, to carry out the

remedial work if required.

6. CONCLUDING REMARKS

Seepage has very bad effect like erosion, piping failure, instability of structures etc.

But we cannot eleminate seepage. We can only direct it by providing easiest path so

that it cannot harm the structure. There are various methods available to measure

the seepage analytically and empirically. We have to choose the suitable method.

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7. REFERENCES

1. Das, B. M., and Sobhan, K. (2014). Principles of Geotechnical Engineering.

2. Terzaghi, K., Peck, R. B., Mesri, G. (1996). Soil Mechanics in Engineering

Practice.

3. Casagrande, A. (1937). Seepage Through Dams.

4. Munson, B., R., Okiishi, T., H., Huebsch, W., W., Rothmayer, A., P., (2013).

Fundamentals Of Fluid Mechanics.

5. Novak, P., Moffat, A., I., B., Nalluri, C., Narayanan, R., (2007). Hydraulic

structures.

6. Garg, S., K., (2013), Irrigation Engineering and Hydraulic Structures.

7. Knappett, J., A., Craig, R., F., (2012). Craig’s Soil Mechanics.

8. Taylor, D., W., (1948). Fundamentals of Soil Mechanics.

9. Arora, K., R., (2004). Soil Mechanics and Foundation Engineering.

10. Venkatramaiah, C., (2006). Geotechnical Engineering.